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LIBRARY 


UNIVERSITY   OF   CALIFORNIA. 

Received  A^i^<^  cJL^i  88  F^ 

A  i  cessions  No .  ^^  ^  ^f/        Shelf  No. 


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OS* 

1 

1    _    _    —     —    _       —          -- 

1 

Digitized  by  the  Internet  Archive 

in  2008  with  funding  from 

IVIicrosoft  Corporation 


http://www.archive.org/details/advancedarithmetOOcalirich 


CALIFORNIA  STATE  SERIES  OF  SCHOOL  TEXT-BOOKS. 


ADVANCED 


AEITHMETIC 


COMPILED    UXDER  THE  DIRECTIOX 


STATE    BOARD    OF    EDUCATION. 


sacramento,  california. 
Printed  at  the  State  Printing  Office. 


Entered  according  to  Act  of  Congress,  in  the  year  1887,  hy  the 

STATE    OF   CALIFORNIA, 

In  the  Office  of  the  Librarian  of  Congress,  at  Washington. 


PREFACE. 


The  State  Board  of  Education  expect  to  make  no  revolution  in 
teaching  the  old  subject  of  Arithmetic,  by  the  issuance  of  a  new 
book.  They  feel,  however,  that  arithmetics  have  been  too  much 
given  to  talking  and  not  enough  to  doing — that  a  student  seldom 
or  never  masters  the  thought  in  a  long  and  minute  explanation. 
He  cannot  understand  it  before  working  the  examples,  and  does 
not  need  it  afterward.  Hence,  the  explanations  in  the  present 
volume  have  been  made  brief,  and  may  be  enlarged  by  the  teacher 
as  the  occasion  demands. 

Let  no  one  despise  the  book  on  account  of  its  small  size,  but  work 
a  class  carefully  through  it,  making  it  familiar  by  frequent  reviews, 
and  observe  the  effect.  We  respectfully  invite  the  candid  criticism 
of  those  who  have  done  this,  that  the  defects  of  the  present  volume 
may  be  remedied  in  the  near  future. 


COKTEE'TS. 


PAGK. 

Notation  and  Numeration 5 

Addition 14 

Subtraction 21 

Multiplication 34 

Division 43 

Factors 63 

Fractions 72 

Short  Methods 115 

Bills 119 

Weights  and  Measures 122,. 

General  Analysis 172 

Proportion 176 

Partnership 178 

Percentage 181 

Profit  and  Loss 185 

Commission 189 

Insurance 194 

Taxes 198 

Stocks 201 

Interest 204 

Partial  Payments 214 

Compound  Interest 216 

Discount 219 

Accounts 221 

Exchange 228 

Average  of  Payments 233 

Average 235 

Powers  and  Roots 237 

Mensuration 246 

Miscellaneous  Problems 257 

Abbreviations 262 

Signs ' 263 

Glossary 264 

Answers 271 

Index 287 


cjlLiforwia.  series. 
ADVANCED 

ARITHMETIC. 


NOTATION  AND  NUMERATION. 

A  single,  whole  thing  is  called  a  unit;  as,  o?ie,  one  apple, 
one  pencil. 

Several  things  taken  together  as  a  whole  may  be  a  unit; 
as,  one  dozen  pencils,  one  pile  of  hooJcs,  one  class  of  hoys. 

A  number  consists  of  one  or  more  units;  as,  one,  one  cent, 
seven,  seven  hooks,  ten  pens. 

Writing  numbers  is  called  NoTATiOiN. 

The  notation  in  common  use  is  the  decimal  notation, 
which  employs  ten  different  characters,  or  figures,  to  form 
all  numbers. 

All  numbers  are  properly  followed  by  a  point  (.).  called 
the  decimal  point ;  as  in  the  table  below.  In  writing  num- 
bers in  a  series  or  in  a  sentence,  the  decimal  point  is  omitted 
to  avoid  confusion  with  the  period. 

The  following  table  gives  the  ten  characters  of  the  deci- 
mal notation  in  the  upper  horizontal  row  and  their  names 
beneath.  Then  follow  their  combinations,  forming  num- 
bers of  two  figures.  Put  this  diagram  on  the  slate  and  fill 
out  completely,  writing  (1)  the  figure  (2)  the  combination 
(3)  the  name. 


CALIFORNIA   SERIES. 


Figure 

Name 


Figure 

Combination 

Name 


Figure 
Combination 

Name 


Figure 

Combination 

Name 


Figure 

Combination 

Name 


Figure 

Combination 

Name 


Figure 
Combination 

Name 


Figure 

Combination 

Name 

Figure 

Combination 

Name 

Figure 

Combination 

Name 


0 
Zero 

1. 

One 

2. 

Two 

3. 
Three 

10. 
Iten 
Ten 

11. 

f  1  ten  \ 

1  1  unit  i 

Eleven 

12. 

f    1  ten    \ 

\  2  units  j 

Twelve 

13. 
(    1  ten    ) 
t  3  units  1 
Thirteen 

20. 

2  tens 
Twenty 

21. 

j  2  tens  ) 

\  1  unit  j 

Twenty-one 

22. 

(  2  tens    ) 

\  2  units  1 

Twenty-two 

30. 
3  tens 
Thirty 

40. 
4  tens 
Forty 

60. 
5  tens 
Fifty 

60. 

6  tens 
Sixty 

70. 

7  tens 
Sevent}" 

80. 
8  tens 
Eighty 

90. 
9  tens 
Ninety 

Observe 


r  Units  form  the  first  figure  at  the  left  of  tlie  decimal 
In  column  i,  the  absence  of  units  is  marled  by  0, 

any  decimal  place  is  always  marked  by  0. 
Hoiv  many  units  make  1  tenf 


ARITHMETIC. 


4. 

Four 


5. 
Five 


14.  15. 

(1  ten     M  f    1  ten 

'(  4  units  )    t  5  units 
Fourteen      Fifteen 


6. 

Six 


Seven 


16. 

f    Iten    ) 

\  G  units  I 

Sixteen 


17. 
I     1  ten     I 
[  7  units  j 
Seventeen 


Eight 


18. 
f    1  ten   ) 
(  8  units  ) 
Eighteen 


9. 
Nine 


19. 

1  ten 

9  units 

Nineteen 


f   1  ten    ) 

\  9  units  j 


point;  tens,  the  second. 

called  nought,  zero,  or  cipher.     The  absence  of  number  in 


94. 

18. 

82. 

72. 

27. 

97. 

13. 

55. 

47. 

29. 

14. 

79. 

85. 

32. 

28. 

44. 

66. 

38. 

51. 

74. 

70. 

90. 

95. 

98. 

49. 

59. 

36. 

37. 

46. 

41. 

20. 

67. 

75. 

88. 

92. 

58. 

50. 

12. 

39. 

19. 

21. 

96. 

8  CALIFORNIA   SERIES. 

EXERCISE   1. 

Draw  a  diagram  like  the  preceding  and  write  the  names 
of  the  following  numbers  and  the  combinations  which 
make  them: 

40. 

77. 

30. 

62. 

71. 

54. 

87. 

EXERCISE  2. 

Write  the  following  combinations  in  figures  and  their 
names  in  words: 

1.  2  tens  4  units,  3  tens  1  unit,  6  tens,  5  tens  3  units. 

2.  8  tens  4  units,  4  tens  2  units,  8  tens,  3  tens  5  units,  6 
tens  8  units,  5  tens  6  units,  6  tens  o  units. 

3.  8  tens  6  units,  5  tens  7  units,  7  tens,  4  tens  5  units,  1 
ten  6  units,  7  tens  3  units,  6  tens  1  unit. 

4.  1  ten  7  units,  9  tens,  6  tens  9  units,  4  tens,  1  ten  8 
units,  7  tens  8  units,  8  tens  1  unit. 

5.  3  tens,  2  tens  6  units,  4  tens  8  units,  9  tens  9  units,  6 
tens  4  units,  2  tens  5  units,  5  tens  8  units. 

6.  4  tens  3  units,  9  tens  1  unit,  8  tens  9  units,  6  tens  5 
units,  2  tens  3  units,  3  tens  4  units,  5  tens  2  units. 

7.  4  tens  1  unit,  7  tens  5  units,  1  ten  7  units,  9  tens  6 
units,  3  tens  3  units,  5  tens  8  units,  1  ten  5  units. 

The  third  figure  at  the  left  of  the  decimal  point  is  called 
hundreds;  thus,  236  is  2  hundreds  3  tens  6  units,  or  tico 
hundred  thirty-six. 

*  How  many  tens  make  1  hundred  ? 

These  three  places  of  figures — units,  tens,  and  hundreds — 
form  the  first  group  of  numbers,  called  units. 


ARITHMETIC.  9 

The  fourth,  fifth,  and  sixth  places  at  tlie  left  of  tlie  deci- 
mal point  form  the  second  group,  called  thousands ;  units, 
tens,  and  hundreds  of  thousands,  respectively. 

The  seventh,  eighth,  and  ninth  places  at  the  left  of  the 
decimal  point,  form  the  third  group,  called  millions ;  units, 
tens,  and  hundreds  of  millions,  respectively. 

The  following  table  shows  the  scheme  for  reading  num- 
bers, or  Numeration.  In  reading,  begin  at  the  left,  read 
each  group,  and  add  the  group  name;  thus,  one  hundred 
twenty-four  sextillion,  seven  hundred  thirty  quintillion,  etc., 
omitting  the  name  of  the  unit  group: 


TABLE. 


m  C  p.  m 

d  O  .2  •  'TJ 

.2  -3  ^  ?2  «^  a  ^ 

^  '-^  r^  O  O  .2  ^ 

■■^  ^  ^  'r^  -^  'B  ^ 

<^  Ej  Bj  T^  •:::)  d  r^ 

02  CT"  CT"  -t5  ^  Pi  qS 

o  o  o  o  o  o  o 


0QO0CQCOCOC/2CCOQ 

Ti  "73  T?  'T3  "T^  "T^  "^  "73 

'T^  xn    Ti  en    'Xi  cotU  zn    Ti  oq'tJ  w    Ti  cc"^  02 

124,730,218,6  9  3,013.978.210,453. 

Note. — The  omission  of  ''and"  between  hundreds  and  tens  is 
the  better  usage,  although  many  writers  and  speakers  still  use  it. 

Suggestion. — Require  oral  exercise  by  the  class  upon  the  preced- 
ing table  until  it  is  familiar  to  all. 

EXERCISE  3. 
Read,  or  write  on  slates  or  blackboard,  in  words: 

1.  208.  4.     727.  7.  7051.  10.     3108. 

2.  523.  5.  4009.  8.     555.  11.     4018. 

3.  1001.  6.     300.  9.     476.  12.  23760. 


10  CALIFORNIA   SERIES. 

13.       1414.         22.       1211.  31.       525.  40.       5729. 


14. 

2007. 

23. 

41407. 

32. 

800. 

41. 

100010. 

15. 

105. 

24. 

270. 

33. 

805. 

42. 

74179. 

16. 

8248. 

25. 

643077. 

34. 

3104. 

43. 

85128. 

17. 

5678. 

26. 

21190. 

35. 

7228. 

44. 

7300. 

18. 

179. 

27. 

758. 

36. 

720. 

45. 

211. 

19. 

24198. 

28. 

7112. 

37. 

5000. 

46. 

2419, 

20. 

179226. 

29. 

987. 

38. 

2726. 

47. 

43200, 

21. 

473. 

30. 

3721. 

39. 

54100. 

48. 

7290. 

EXERCISE  4. 
Write  the  following  in  figures,  to  be  read  in  the  class: 

1.  Five  hundred  seventy-two,  one  thousand  seventeen, 
five  thousand  ninety,  four  hundred  sixty-four,  twenty-four 
thousand  eight,  three  hundred  forty-six,  nine  thousand 
ninety-nine. 

How  many  groups  are  employed  in  writing  the  first  number? 
How  many  in  writing  the  second?  The  third?  What  places  are 
vacant  in  each  group?    They  should  be  occupied  by  zero. 

2.  Eleven  thousand  seven  hundred  eighty-five,  seventeen 
thousand  twenty-nine,  eight  hundred  eight,  three  thousand 
fifteen,  eighteen  thousand  thirty,  twenty-five  thousand  four 
hundred,  seven  hundred  six. 

3.  Forty  thousand  nine  hundred  three,  sixty-one  thou- 
sand three  hundred  thirty-three,  one  hundred  four  thousand 
twenty,  seven  thousand  forty-six,  eight  hundred  eighty- 
eight  thousand  eight,  nine  hundred  sixty-nine,  two  thou- 
sand four  hundred  thirteen. 

4.  Fourteen  thousand  seven  hundred  forty-five,  two 
hundred  fifty-one  thousand  one  hundred  sixteen,  thirty- 
four  thousand  one  hundred  eleven,  five  thousand  sixty-six, 
thirty-one  thousand  nine  hundred  fifty-two,  eighty-two 
thousand  three  hundred  twelve. 

5.  Nineteen  thousand  five  hundred,  seven  thousand  four 
hundred  twenty-three,  six  hundred  nine,  six  hundred  nine 


ARITHMETIC.  11 

thousand,  six  thousand  nine,  fifty-nine  thousand  five,  five 
thousand  nine  hundred  five. 

6.  Three  thousand  thirteen,  three  hundred  thirteen, 
three  hundred  thousand  thirteen,  thirty  thousand  thirteen, 
eight  hundred  eighty-one,  eight  thousand  eighty-one,  eighty 
thousand  eighty-one. 

EXERCISE   5. 

Write,  on  your  slates,  through  the  group  of  milhons,  a 
table  like  that  on  page  9,  and  place  under  it  in  vertical  col- 
umn 20  numbers  of  your  own  selection,  containing  from 
3  to  9  places  each,  for  reading  and  dictation  in  the  class. 

EXERCISE  6. 
Read  the  numbers  under  Exercises  22  and  24. 

EXERCISE  7. 
Write  the  following  in  figures: 

1.  5  thousand  2  hundred  10,  24  thousand  6  hundred  3, 
11  thousand  29,  7  hundred  63,  16  thousand  8  hundred,  4 
hundred  44. 

2.  123  thousand  123,  14  hundred  14,  73  thousand  5 
hundred  8, 17  hundred,  141  thousand,  3  million  3  thousand 
3  hundred  3. 

3.  7  thousand  7,  7  hundred  7,  7  million  7  thousand  7, 
7  million  7,  13  hundred  30,  13  thousand  30. 

4.  115  thousand  7  hundred  74,  10  hundred  10,  10  thou- 
sand 10,  1  thousand  10,  5  hundred  91,  8  thousand  4  hun- 
dred 20. 

5.  404  thousand  44,  23  thousand  213,  180  thousand  180, 
47  thousand  474,  3  thousand  206,  eighty-one. 

6.  826  thousand  013,  15  thousand  411,  111  thousand 
111,  400  thousand  400,  328  thousand  910,  50  thousand  50. 

7.  501  thousand  107,  55  thousand  76,  28  thousand  1. 

8.  101  thousand  10,  101  million  1  thousand  6. 

9.  110  thousand  11,  20  million  11  thousand  11. 


12 


CALIFORNIA   SERIES. 


EXERCISE    8. 

Dictation  exercise  by  the  class,  each  giving  his  own  num- 
bers without  reference  to  book,  slate,  or  paper.  Repeat  this 
exercise  until  the  class  dictate  and  write  rapidly. 

EXERCISE    9. 

Place  the  following  in  tabular  form,  as  in  Exercise  5,  for 
reading  in  the  class: 

.    224368192,  1724261,  2004101,  7264180,  2010194,  3762108, 
23101,   47266,    4004,    20801,    76001,    2108,  17007,  100100. 

Another  notation,  called  the  Roman  notation,  is  some- 
times used  for  writing  dates,  headings  of  chapters,  and  the 
like;  but  it  is  too  cumbrous  for  ordinary  computations. 
The  Roman  notation  eniploys  seven  capital  letters,  with 
their  combinations,  to  represent  numbers,  viz.: 

I        V      X      L  C  D  M 

One,    five,    ten,    fifty,    one  hundred,   five  hundred,  one  thousand. 
1        5      10      50  100  500  1000 

The  following  table  shows  the  method  of  combining: 


I  . 

one. 

VII    . 

.     seven. 

LX     .     .     .    sixty. 

II   . 

two. 

VIII    . 

eight. 

XC     .     .      ninety. 

Ill   . 

.  three. 

IX    . 

.  nine. 

XL     .     .     .    forty. 

IV     . 

.     four. 

X    . 

.    ten. 

L     .     .     .     fifty. 

V    . 

five. 

XI    . 

eleven. 

C  .  one  hundred. 

VI    . 

.     .  six. 

XX    . 

twenty. 

D  .  five  hundred. 

M  one 

thousand. 

M  one  n 

lillion. 

Observe  ^ 


'  Repeating  a  letter  repeats  its  value. 

If  a  letter  of  smaller  value  precedes  one  of  larger, 
the  difference  of  their  values  is  indicated;  if  the 
reverse,  the  sum. 

A  dash  ( — )  above  a  letter  indicates  so  many  thou- 
sand; thus,  L  =  fifty  thousand. 


ARITHMETIC. 


13 


EXERCISE  10. 
Write  in  Roman  notation: 

8,  14,  27,  144,  1875,  599,  1620,  35,  178,  83,  124000,  753, 
16,  222,  1888,  7,  12,  79. 

EXERCISE   11. 

Read  the  following  numbers: 

XIX,  XXIX,  XXXVI,  CCCI,  ex,  DCLII^  CDXIV, 
MDLXXXIV,  MDCCCLXXXVI,  CXLVII,  MC,  XCIX, 
CCCXXV,  LXXII,  DIV,  MCCXVIII,  CXI,  DCCXLVII, 
MDCCLXXXIX,  MCDXCII,  CL,  CCXV. 

EXERCISE   12. 
Prepare  3  columns  on  your  slate  as  follows: 

First  column,  10  numbers  written  in  words; 

Second  column,  the  same  numbers  in  decimal  notation; 

Third  column,  the  same  in  Roman  notation. 

Model : 


No. 

"Words. 

Decimal  Xotatitin. 

Eoman  Notation. 

1. 

2. 

3. 
4. 
5. 
G. 

7. 

8. 

9. 

10. 

Twenty-five. 

25. 

XXY. 

14  CALIFORNIA   SERIES. 


ADDITION. 

If  you  have  8  apples  and  a  schoolmate  gives  you  5  more, 
how  many  will  you  have  ? 

The  process  of  putting  together  two  or  more  numbers  of 
the  same  kind  into  one  is  called.  Addition. 

The  result  obtained  is  called  the  sum  or  amount. 

The  sign  (-|-),  called  plus  or  and,  is  used  to  indicate  ad- 
dition.    Thus, 

8-\-5=^lo  is  read  8  plus  5  equals  13^  or  8  and  5  are  13. 

Suggestion. — With  beginners  ''and"  is  preferable  to  plus. 

EXERCISE   13.    (Oral.) 

To  THE  Teacher. — Give  pupils  pebbles,  beans,  peas;  or,  better, 
pasteboard  cut  into  strips  }q  in.  wide  and  3  in.  long,  to  find  out  the 
results  by  going  through  with  the  combinations.  Drill  on  the  fol- 
lowing until  the  pupil  recognizes  at  sight  the  sum  of  each  pair : 

123132214353526428 


2 

1 

1 

3 

2 

3 

2 

4 

1 

3 

2 

6 

5 

5 

3 

2 

6 

3 

6 

5 

7 

8 

6 

9 

4 

5 

3 

1 

7 

9 

8 

5 

2 

6 

6 

4 

5 

9 

6 

8 

4 

3 

4 

5 

6 

2 

9 

7 

3 

9 

8 

6 

9 

24772736529813 
84974542176597 

EXERCISE  14.    (Written.) 
Write  each  of  the  pairs  and  their  sum,  in  Exercise  13, 
horizontally,  using  the  signs  (  +  )  and  (=);  thus,  1-^2=3. 
Bring  to  the  class  to  read. 

EXERCISE    15.    (Oral.) 
Add  these  columns,  taking  the  figures  in  pairs,  and  call- 


ARITHMETIC,  I   •"  J5 

ing  only  the  sums  of  the  pairs  ;  thus,  in  the  first; -?<?,  15)  abo 
add  across  the  page  from  left  to  right  as  indicated  by  the 
sign(+)  : 

1+3+4+3+8+7+8+7+5-1-8=:? 
4+2+3+9+4+6+5+8+5+1=? 
.7+5+7+4+3+9+8+5+6+4=? 
3+6+9+6+1+5+9+4+3+9=? 

7+9+7+8+2+2+9+6+2+2=? 
1+1+7+8+8+9+9+9+1+6=? 
9+6+3+5+4+2+2+3+8+4=? 

3+6+8+7+4+7+5+3+6+7=? 

EXERCISE  16.  (Oral.) 

Extend  Exercise  15  by  beginning  with  the  second  figure 
on  the  left  in  each  horizontal  row  and  adding  through  the 
row.     Then  begin  with  the  third,  and  so  on. 

EXERCISE  17.    (Written.) 
Copy  the  following,  fill  out  as  indicated,  and  write  results: 

1.  2.                     3.                    5.  8. 

2+5=  3+7=           7+8=           3+9=  9+9= 

12+5=  13+7=  17+8=  13+9=  19+9= 

22+5=  23+7=  27+8=  and  so  on.  and  so  on. 

32+5=  33+7=  37+8=              6.  9. 

42+5=  43+7=  and  so  on.         6+6=  7+7= 

52+5=  53+7=              4.  16+6=  17+7= 

62+5=  63+7=           7+6=  and  soon,  and  soon. 

72+5=  73+7=  17+6=              7.  10. 

82+5=  83+7=  27+6=           4+7=  1+9= 

92+5=  93+7=:  and  so  on.  and  so  on.  and  so  on. 

EXERCISE   18.    (Oral.) 

1.  Begin  with  0  and  add  by  2's  to  50;  thus,  0,  ^,  4,  6, 
etc.     Do  the  same,  beginning  with  1,  to  51;  thus^  I,  o,  Oj  etc. 


16  CALIFORNIA   SERIES. 

2.  Add  by  3's  from  0  to  51;    from  50  to  98;   from  2  to 
50;  from  51  to  99. 

3.  Add  by  4's  from  0  to  52;    from  51  to  99;    from  2  to 
50;  from  52  to  96. 

4.  Add  by  5's  from  0  to  100;  from  1  to  101;  from  2  to 
102;  from  3  to  103;  from  4  to  104. 

EXERCISE  19.    (Written.) 
Write  as  in  Exercise  17: 

1.                    2.                    3.  4.  5. 

4+9=           3+8=          9+7=  8+8=  8+5= 

14+9=  13+8=  19+7=  18+8=  18+5= 

24+9=  23+8=  29+7=  28+8=  28+5= 

and  so  on.  and  so  on.  and  so  on.  and  so  on.  and  so  on. 

6.  7.  8.  9.  10. 

5_|-8=  9+2=  2+9=  6+5=  5+6= 

15+8=  19+2=  12+9=  16+5=  15+6= 

25+8=  29+2=  22+9=  26+5=  25+6= 

and  so  on.  and  so  on.  and  so  on.  and  so  on.  and  so  on. 

Suggestion. — Let  the  teacher  give  further  oral  work  of  the  same 
kind. 

EXERCISE  20.    (Oral.) 

1.  Add  by  6's  from  0  to  60;    from  52  to  100;    from  1  to 
61;  from  43  to  103;  from  5  to  65. 

2.  Add  by  7's  from  0  to  70;   from  3  to  73;  from  32  to 
102;  from  16  to  86;  from  31  to  101. 

3.  Add  by  8's  from  0  to  80;    from  21  to  101;    from  5  to 
85;  from  17  to  97;  from  6  to  86;  from  43  to  123. 

4.  Add  by  9's  from  0  to  90;    from  2  to  101;    from  5  to 
104;  from  13  to  103;  from  8  to  98;  from  7  to  106. 

EXERCISE  21.    (Oral.) 
(1)  Begin  with  the  bottom  of  each  colmnn  and  add  up- 
ward in  pairs  (2)  begin  with  the  top  and  add  downward 
in  pairs  (3)  add  across  from  left  to  right: 


ARITHMETIC.  17 

7_|_5_|-3_^6+7+2+9+3+8H-4=? 
2+2+3+8+4+9+2+9+1+6=? 
9+9+9+1+6+2+9+6+2+2==? 
2+9+6+2+2+4+3+9+3+6=? 
4_^3_|_9_j_3-|_6+5+6+4+9+6=? 
5+6+4+9+6+8+5+1+1+1=? 
8+5+1+1+1+7+5+8+7+9=? 

5+3+9+3+6+8+7+4+7+5=? 
8+6+4+9+6+3+5+4+2+2=? 
7+5+1+1+1+7+8+8+9+9=? 
3+5+8+7+9+7+8+2+2+9=? 
7+6+9+6+1+5+9+5+3+6=? 
4+5+7+4+3+9+8+8+7+5=? 
1+2+3+9+4+6+7+1+4+2=? 
4+3+4+3+8+7+8+1+1+3=? 

1+4+7+3+3+2+5+6+5+5=? 
3+2+5+6+4+3+7+9+8+1=? 
4+3+7+9+3+9+4+6+7+1=? 
3+9+4+6+8+4+3+1+9+1=? 
8+4+3+1+7+6+9+5+7+7=? 
7_|_6+9+5+8+7+8+9+8+8=? 
8+7+8+9+1+1+8+5+2+8=? 
1+1+8+5+7+8+5+4+2+9=? 
7+8+5+  4+5+5+6+3+9+9=? 

This  exercise  may  be  extended  by  beginning  at  any 
intermediate  point  and  adding  onward. 

To  write  and  add  numbers  of  two  or  more  figures. 

Suggestion. — Before  allowing  the  pupils  to  study  this  or  any  ^um- 
lur  explanation ,  the  teacher  should  take  the  work  orally  with  them 
and  let  them  make  as  many  of  the  suggestions  as  they  can. 

Find  the  sum  of  327,  48,  and  452. 
2— A 


FULL  WORK 

327 

48 

452 

17 

110 

700 

18  CALIFORNIA   SERIES. 

Explanation. — We  cannot  add  7  pencils  and  6  pens, 
because  they  are  unlike.  Likewise,  we  cannot  add 
tens  to  units.  Therefore,  write  units  under  units,  tens 
under  tens,  etc.  The  sum  of  the  units  is  17;  of  the 
tens,  11;  of  the  hundreds,  7.  Adding  these  results 
gives  827. 

Why  is  5;ero  (0)  placed  after  11? 
Why  are  two  zeros  (00)  placed  after  7? 

827 

But  17  units  are  1  ten  7  units.     As  the  1  ten  belongs 

CONTRACTED,  in  the  tens'  column,  we  may  write  only  the  7  units, 

3  2-7        as  in  the  contracted  operation,  and  add  the  1  ten  to  the 

43        tens'  column;  thus,  1,  6,  10,  12.     Again,  12  tens  are  1 

^  r  9        hundred,  2  tens.     As  before,  write,  in  the  result,  only 

the  2  tens  and  add  the  1  hundred  to  the  column  of 

8  2  7        hundreds. 

Test^  or  prove,  the  correctness  of  the  ivork  hy  adding  down- 
ward. 

EXERCISE  22.    (Written.) 

Write  in  columns  properly,  add,  and  test  the  work: 

1.  424,  236,  38,  120.         11.  1234,  4321,  1324,  4231. 

2.  34,  108,  246,  5.  12.  3579,  9753,  3795,  9573. 

3.  402,  1728,  526,  100.       13.  908,  7098,  9708,  987. 

4.  3756,  11,  153,  4005.       14.  7890,  798,  8790,  809. 

5.  271,  109,  9019,  49.        15.  4796,  7694,  976,  479. 

6.  7310,  101,  476,  1203,  45.    16.  3251,  1523,  5237,  8. 

7.  423, 13, 9,  237,  2314, 103.     17.  487,  9217,  1499,  7. 

8.  19,  500,  275,  2406,  2728,  2010.  18.  534,  434,  898,  10. 

9.  9019,  428, 1300,  23,  99,  3003.  19.  921,  651,  1397,  14. 
10.  1314,  810,  278,  4130,  44, 176.  20.  1455,  1085,  95,  117. 

EXERCISE  23. 

Add  the  columns  of  figures  in  Exercise  1,  and  test.  Add 
the  same  across  the  page. 

Write  each  example  of  Exercise  2  in  column,  add,  and 
test. 


ARITHMETIC.  19 

EXERCISE  24. 

Add  and  prove,  in  columns  and  in  rows: 

1.  2.  3.  4.         5. 

15.  4298+1029+  428+7296+     49=? 

16.  376+     76+5001+     98+1311=? 

17.  107+  237+     19+  402+  205=? 

18.  25+4196+1279+     13+     15=? 

19.  3178+  703+  499+  720+3146=? 

6.  7.  8.  9.         10. 

20.  9+  207+  575+3209+1712=? 

21.  79+3426+     82+  729+5726=? 

22.  4327+  127+  426+     48+  209=? 

23.  214+5728+1350+7216+8702=? 

24.  903+4019+  407+  590+  435=? 

11.             12.             13.  14. 

25.  375409+  72496+  718409+  419009=? 

26.  23216+570203+     20171+  21060=? 

27.  25100+  30206+  376219+  1199=? 

28.  5196+175410+4211010+  519257=? 

29.  576206+  76228  +  5176159+4219219=? 

30.  61070+481112+  172105+  728400=? 

EXERCISE  25. 
Add  the  columns  of  figures  in  Exercise  3,  and  test.     Add 
the  same  across  the  page.     AVrite  each  example,  Exercises 
4  and  7,  in  columns,  add,  and  test. 

EXERCISE  26. 
Write  10  examples  of  your  own,  of  10  numbers  each, 
perform,  and  prove,  and  bring  into  the  class  to  dictate  to 
the  others  for  board-work. 

EXERCISE  27. 
Dictate  numbers  of  your  own,  without  reference  to  book, 


20  CALIFORNIA   SERIES. 

slate,  or  paper,  for  the  other  members  of  the  class  to  per- 
form.    Repeat  the  exercise  until  each  dictates  rapidly. 

Accomitants  and  business  men,  by  constant  practice,  add 
two  or  even  three  columns  at  once  with  great  rapidity. 

To  add  two  columns,  it  is  customary  to  add  the  tens  first 
and  then  the  units  in  each  successive  number. 


24 
57 


Thus,  3  tens  4  units +  4  tens  (  =  7  tens)  6  units  =  8  tens  0 
units, +5  tens  (  =  13  tens)  7  units  =  13  tens  7  units, +2  tens 
4  6  (  =  15  tens)  4  units  =  1G  tens  1  unit.  In  reading  omit  parts 
3  4  in  parenthesis  and  the  words,  tens  and  units;  thus,  5,  4; 
8,  0;         13,  7;         16,  1. 


161 


24  88 

17  40 

50  26 

75  23 

29  81 

31  44 


(Written  or  Oral.) 
Copy  and  fill  out,  or  read,  as  in  the  first  example: 
30^20=50=5  tens. 


19 

5721 

3333 

12 

4804 

3214 

4141 

6789 

1818 

2AU 

7117 

7642 

7236 

2104 

4261 

4004 

5016 

1781 

40+30= 

70+50= 

100+20= 

120+60= 

40+20= 

90+40= 

100+30= 

140+70= 

70+20= 

80+20= 

110+40= 

130+80= 

80+10= 

50+40= 

110+70= 

110+60= 

50+30= 

30+20= 

130+40= 

150+90= 

60+40= 

60+30= 

150+70= 

100+40= 

90+70= 

80+40= 

110+50= 

140+50= 

70+40= 

40+10= 

120+70= 

160+80= 

80+50= 

90+60= 

100+50= 

170+90= 

30+10= 

60+20= 

140+60= 

150+80= 

Observe    I  ^^^^^9  -^^'^  '^  ^^'^  (/wes  lO^s,  as  adding  units  to 
{      units  gives  units. 


ARITHMETIC. 


21 


SUBTRACTION. 


Copy  the  following  on  your  slates,  and  put  in  place  of 
each  blank  the  number  that  you  must  add  to  the  one  above 
the  line  to  make  the  One  below: 


2  pencils 
pencils 


4  pens 
pens 


5  apples 
apples 


9  books 
books 


marbles 
marbles 


3  pms 
pins 


9  apples 


17  l)Ook^ 


10  marbles 


9  pins 


7  pencils     11  pens 

Copy  the  following,  also.  Place  below  each  line  the  num- 
ber that  will  be  left,  if  you  take  away  the  lower  from  the 
upper  number: 

11  pens 


7  pencils 
5  pencils 


/  pens 


9  apples 
4  apples 


17  books 
8  books 


16  marbles 
8  marbles 


9  pins 
6  pins 


pencils 


pens 


apples 


books 


marbles 


pms 


Compare  these  two  exercises.  What  did  you  do  in  the 
first  ?  In  the  second  ?  Since  5  put  with  2  makes  7,  5  taken 
from  7  will  leave  2. 

The  process  of  taking  one  number  from  another  of  the 
same  kind  is  called  Subtraction. 

The  number  from  which  Ave  take  is  called  the  minuend. 

The  number  taken  away  is  called  the  subtrahend. 

The  number  left  is  called  the  difference  or  remainder. 

Pick  out  each  in  the  second  iiart  you  copied  above. 

The  sign  ( — ),  called  minus  or  less,  is  used  to  indicate  sub- 
traction ;  thus, 

7 — 5-—2  is  read  7  minus  5  equals  S,  or  7  less  5  are  2. 

Suggestion. — With  beginners,  leas,  is  to  be  preferred  to  minus. 
EXERCISE   28.   (Written.) 

Place  in  each  blank  the  number  that  must  be  added  to 
the  number  above  the  line  to  make  the  one  below: 


22  CALIFORNIA    SERIES. 

47215      675848627 

12  16171311181413171518141219 
591       273     10       6     13       9258 

181316^15  1715121718151211 
3     13       5     12     14       5       3       9       6       7     10       4     12 

1816nr71816i41113181914T6 

EXERCISE   29.    (Oral.) 

Perform  Exercise  18  backward.  That  is,  in  (1)  begin 
with  50  and  take  away  2  each  time;  thus,  50,  48 ^  4^,  etc. 
Then  begin  with  51;  51^  ^9,  ^7,  etc. 

Then  in  (2)  begin  with  51  and  subtract  by  3's;  and  so  on. 

Repeat  until  all  subtract  readily. 

EXERCISE  30.    (Written.) 

Copy  the  following,  fill  out  as  indicated,  and  write  the 
remainders: 


1. 

2. 

3. 

4. 

5 

. 

i  — 

-5= 

10- 

-7= 

15- 

-8= 

13- 

-6--= 

12- 

-9= 

17- 

-5= 

20- 

-7= 

25- 

-8= 

23- 

-6= 

22- 

-9= 

27- 

-5= 

30- 

-7= 

35- 

-8= 

33- 

-6-= 

32- 

-9= 

37- 

-5= 

40- 

—  /  = 

45- 

-8= 

43- 

-6= 

42- 

-9= 

and 

soon 

and 

so  on 

and 

so  on 

and 

.  so  on 

and  so  on 

to  9 

7—5. 

to  100-7. 

to  9 

5—8. 

toS 

>3— 6. 

to  92 

1—9. 

e 

>. 

7. 

B. 

9. 

10. 

12- 

-6= 

11- 

—  /  = 

18- 

-9= 

14- 

r^ 

10- 

-9= 

22- 

-6= 

21- 

—  /  = 

28- 

-9= 

24- 

—  / : 

20- 

-9== 

32- 

-6= 

31- 

_'7 

38- 

-9= 

34- 

-7= 

30- 

-9^ 

42- 

-6= 

41- 

_'7 . 

48- 

-9= 

44- 

-7= 

40- 

-9= 

and 

soon. 

and 

so  on. 

and 

so  on. 

and 

so  on. 

and  so  on. 

Compare  this  work  with  that  of  Exercise  17, 


ARITHMETIC.  23 

EXERCISE  31.    (Oral.) 

Subtract  the  lower  number  from  the  upper: 

20       32        31        25       21        23        33        22        16        43 
9535       11         89893 


39 

18 

26 

59 

47 

64 

71 

74 

41 

55 

4 

1 

0 

8 

7  • 

5 

6 

7 

5 

6 

72 

85 

91 

67 

54 

35 

46 

69 

56 

44 

8 

5 

10 

8 

8 

5 

o 

O 

8 

77 

86 

98 

83 

90 

49 

57 

43 

38 

25 

9 

4 

3 

1 

8 

J 

9 

9 

9 

40 

52 

71 

50 

41 

53 

61 

84 

91 

50 

2 

3 

5 

5 

2 

4 

5 

3 

4 

70 

23 

87 

85 

91 

41 

53 

28 

93 

96 

6 

4 

5 

I 

8 

6 

4 

^ 
i 

5 

9 

EXERCISE  32.    CWritten.) 
Fill  out  as  directed  in  Exercise  30: 


1. 

2. 

3. 

4. 

5. 

13-4= 

11—3= 

11—9= 

16—9= 

16-8= 

23-4= 

21—3= 

21—9= 

26-9= 

26-8= 

and  so  on 

and  so  on 

ana  so  on 

and  so  on 

and  so  on 

to  93—4. 

to  91—3. 

to  91—9. 

to  96—9. 

to  96—8. 

6. 

7. 

8. 

9. 

10. 

13-8= 

14—8= 

10—6= 

11—6= 

17-8= 

23—8= 

24—8= 

20-6= 

21—6= 

27—8= 

and  so  on 

and  so  on 

and  so  on 

and  so  on 

and  so  on 

to  93—8. 

to  94—8. 

to  90  6. 

to  91—6. 

to  97—8, 

EXERCISE  33.  (0 

RAL.) 

Perform   the  work  of  Exercise   20  backwards,  as    you 
were  directed  in  Exercise  29. 


24  CALIFORNIA    SERIES. 

EXERCISE  34.    (Written  or  Oral.) 
Copy  and  fill  out,  or  read,  as  in  the  first  example: 
30—30=10--=!  ten. 


40—30=. 

70—50= 

100—20= 

120—60: 

40—20= 

90—40= 

100  30= 

140—70: 

70  20= 

80—20= 

110—40= 

130—80: 

80—10= 

50  40= 

■  110—70= 

110—60: 

50—30= 

30—20= 

130  40= 

150—90: 

60—40= 

60—30= 

150  70= 

100—40 

90  70= 

80—40= 

110—50= 

140-50: 

70—40= 

40  10= 

120—70= 

160—80 

80—50= 

90—60= 

100—50= 

170—90: 

30—10= 

60—50= 

140—60= 

150—80: 

^,  (  Suhtractmq  10  s  from  10  s  leaves  10  s,  as  subtract- 

Observe    <       ,  . '^  ^      ^      .     -,  . 

(      mg  units  jrom  units  leaves  units. 

EXERCISE  35.    (Oral.) 

1.  15—7+4—10+9+6—10+1—4+9+3+10—5=? 

2.  5+9—1+7—3—5—2+11+4—9+3—10+1=? 

3.  1+9—10+5+3—7+10—11+3—2+8+9—8=? 

4.  43— 5+2— 10— 9+4— 10+3— 7+3— 9+5— 10=? 

5.  17+5—11—1+8—16+4—6+11+9—3—2+5=? 

6.  44+6—10+1—9+8—10+3—8+2—9+2—11=? 

7.  90—20+  5—10—  5+1—10+ 2—9+3—8—9—10=? 

8.  1+17+2—9—10+7+8—11+5—10+0+1+4=? 

9.  3+9—11+10+4—3+7+1—7+2—8+2+1=? 
10.  2+6—7+10+4+3—9—1+3+2—6—7=? 

To  subtract  numbers  of  two  or  more  figures. 

Take  416  from  829. 


OPERATION 

829 


Explanation. — Write  the  subtrahend  under  the 
minuend,  units  under  units,  etc.,  as  in  Addition, 
4  1  G  and  for  the  same  reason.     (What  reason?)     Begin 

^\'^  with  units. 


ARITHMETIC.  25 

EXERCISE   36.    (Written.) 

1.  178—134=  6.  447—336=  11.  4391—1290= 

2.  495_274=  7.  678—567=  12.  7448—5346= 

3.  982—471=  8.  595—494=  13.  8254—3223= 

4.  778—545=  9.  309—207=  14.  9725—2501= 

5.  904—503=  10.  828—721=  15.  3486—1376= 

Take  479  from  627. 
FULL  OPERATION.  EXPLANATION. — "\Ye  Can  not  take  9 

500+14  0+ 1  7=6  2  7  units  from  7  units.  Take  away  1  ten 
400+    70+    9^479        from  the  2  tens  of  the  minuend  and 

put  it  with  the   7  units,  making  17 

100+  40+  8=148  units.  9  units  from  17  units  leave  8 
units,  which  we  write  below  in  the  units  column.  We  can  not 
take  7  tens  from  1  ten  (left  in  the  minuend);  hence  take  1  hundred 
from  the  6  hundred  in  the  minuend  and  put  it  with  the  1  ten,  mak- 
ing 11  tens.  7  tens  from  11  tens  leave  4  tens.  4  hundreds  from  5 
hundreds  leave  1  hundred. 

Test  by  adding  the  remainder  and  subtrahend;  the  result 
should  be  the  minuend. 

EXERCISE  37.    (Written.) 
Write  properly,  find  the  differences,  and  prove  : 

1.  738  and  542.  4.  500  and  430.         7.  1247  and  8146. 

2.  239  and  410.  5.  378  and  909.         8.  598  and  399. 

3.  5786  and  4310.       6.  246  and  725.         9.  979  and  451. 

Sometimes,  when  our  minuend  figure  is  too  small,  it  hap- 
pens that  the  next  minuend  figure  is  0,  or  nothing  to  take 
from.  In  such  a  case  go  to  the  first  minuend  figure,  not  0, 
to  the  left,  and  reduce  down.     Thus, 

Subtract  2378  from  5005. 

_„^„  .-„T^xT  Explanation. — AVe  can  not  take  8  from  5,  and  the 

4  9  9  15  next  two  minuend  figures  are  O's.    We,  therefore,  take 

500  5  1  thousand  from  the  5  thousands,  leaving  4  thousands, 

o  o  --  o  as  shown  by  the  small  figure  above.     1  thousand  is 

10  hundreds.     Again,  take  1  hundred  from  the  10  hun- 

2  6  2  7  dreds,  leaving  9  hundreds,  as  shown  above.     1  hun- 


26  CALIFORNIA   SERIES. 

dred  is  10  tens.  Take  1  ten  from  10  tens,  leaving  9  tens.  1  ten  5 
units  are  15  units.  Now  subtract  tlie  subtraliend  figures  from  the 
small  figures  above  the  minuend. 

EXERCISE  38.    (Oral  and  Written.) 

Take  the  lower  from  the  upper  numbers;  also  subtract 
as  indicated  by  the  sign  ( — ) ;  prove  your  work. 

1.        2.  3.         4.  5.        6. 

13.  800—143=?         15.  1467—300=?         17.  671—420=? 

14.  75—  29=?         16.     229—  85=?         18.  176--  89=? 

7.        8.  9.         10.  11.      12. 

19.  1100—240=?       21.  1728—1128=?       23.  990—871=? 

20.  73—  19=?       22.     411—  301=?        24.  747—  75=? 


EXERCISE    39.    (Written.) 

Write  and  find  the  difference  between  the  first  two  num- 
bers of  each  example  in  Exercise  22 ;  prove  your  w^ork. 
Thus 

^'236  '    34 

Do  the  same  with  the  last  two  numbers  of  each  example. 

EXERCISE  40.    (Oral  or  Written.) 

Find  the  difference  between  each  number,  except  the  last, 
and  the  next  one  below  it  in  examples  1  to  14,  Exercise  24. 
Finish  with  the  numbers  of  Example  1,  then  take  those  of 
Example  2,  and  so  on.  Number  your  examples  as  you 
write  them. 

EXERCISE  41. 

Find  the  difference  between  each  number,  of  the  first 
two,  and  the  second  number  below  it  in  examples  1  to  14, 
Exercise  24;  between  the  first  number  and  the  third  num- 
ber below  it.     Work  in  the  same  order  as  in  Exercise  40. 

EXERCISE  42. 

Find  the  difiference  between  the  first  number  of  Example 


ARITHMETIC.  27 

1,  Exercise  22,  and  the  first  number  of  each  of  tlie  other 
examples.     Tlius, 

1.  4^4—34;  2.  424—402;  3.  3756—424;  and  so  on. 

Difference  between  the  second  number  of  Example  1  and 
the  first  number  of  each  of  the  other  examples  ;  the  third 
number  of  Example  1  and  the  first  number  of  each  of  the 
other  examples  ;  the  fourth  number  of  Example  1  and  the 
first  number  of  each  of  the  other  examples. 

EXERCISE  43. 

Find  the  difference  between  each  number,  except  the  last, 
and  the  next  number  to  the  right  in  examples  15  to  30, 
Exercise  24.  Finish  with  each  line  before  proceeding  to 
the  next. 

EXERCISE  44. 

Difference  between  each  number,  of  the  first  two,  and 
the  second  number  to  the  right  in  examples  15  to  30,  Exer- 
cise 24  ;  between  the  first  number  and  the  third  number  to 
the  right.     Work  in  the  same  order  as  in  Exercise  43. 

EXERCISE  45. 

Write,  perform,  and  prove  20  examples  of  your  own  in 
Subtraction.     Bring  to  the  class  to  dictate  to  the  others. 

EXERCISE  46. 

Dictate,  without  writing  them  and  without  help,  numbers 
of  your  own,  to  the  others  of  your  class. 


28  CALIFORNIA   SERIES. 


PRACTICAL  WORK  IN  ADDITION  AND  SUBTRACTION. 

All  examples  in  Addition  and  Subtraction  may  be  re- 
duced to  one  of  the  following  general  forms : 

General  (  A.— Find  the  sum  of  327,  48,  and  452. 
Forms.  \  B.— Find  the  difference  between  479  and  627. 

Illustration  1. — A  man  has  256  trees  in  one  orchard  and 
375  in  another  ;  how  many  has  he  in  both? 

We  are  to  put  together,  or  add,  the  trees  in  both  orchards  ; 
hence,  the  general  form  for  the  example  is: 

A.  Find  the  sum  of  256  and  375. 

Illustration  2. — A  man  having  324  oranges  sold  108  of 
them  ;  how  many  had  he  left? 

We  are  to  take  away  the  number  of  oranges  sold  from  the 
whole  number  he  had  ;  hence,  the  general  form  for  this  ex- 
ample is: 

B.  Find  the  difference  between  324  and  108. 

EXERCISE  47. 
Think  of  each  example  carefully,  find  out  what  is  asked, 
and  then  write  the  general  form  for  each  of  the  first  20  ex- 
amples below: 

1.  A  man  had  a  ranch  of  4750  acres,  from  which  he  sold 
1287  acres ;  how  many  acres  had  he  left? 

2.  A  man  sets  oat  an  orchard  of  156  pear  trees,  273  apri- 
cot trees,  195  peach  trees,  390  apple  trees,  and  312  almond 
trees  ;  how  many  trees  were  in  the  orchard  ? 

3.  A  boy  saves  $83  the  first  year  after  leaving  school,  and 
$147  the  second;  how  much  does  he  save  in  both? 

4.  Two  men  walk  a  three  days'  race.  One  travels  263 
miles;  the  other,  197.  How  many  more  miles  does  one 
walk  than  the  other? 

5.  There  are  31  days  in  January,  28  in  February,  31  in 


ARITHMETIC.  29 

March,  30  in  April,  31  in  May,  30  in  June,  31  in  July,  31  in 
August,  30  in  September,  31  in  October,  30  in  November, 
and  31  in  December.  How  many  days  are  there  in  the 
whole  year? 

6.  I  paid  $2500  for  a  house,  $350  for  a  horse  and  buggy, 
$65  for  a  cow,  $119  for  furniture,  and  $47  for  groceries; 
what  did  I  pay  for  all? 

7.  The  number  of  people  in  Sacramento  in  1870  was 
16283;  in  1880,  21420.  How  many  more  people  were  in 
Sacramento  in  1880  than  in  1870? 

8.  Gen.  Grant  was  born  in  1822  and  died  in  1885;  how 
old  was  he  when  he  died  ? 

9.  The  Mississippi  River  is  2816  miles  long;  the  Missouri, 
3047.     Which  is  the  longer  and  how  much? 

10.  In  1882,  Alameda  County  cast  4617  votes  for  George 
Stoneman  for  governor;  Los  Angeles,  3943;  Sacramento, 
3248;  San  Francisco,  24257;  Santa  Clara,  3308.  How  many 
votes  did  these  5  counties  cast  for  Mr.  Stoneman? 

11.  How  many  more  votes  were  cast  by  San  Francisco 
County  than  by  the  other  4  counties  put  together? 

12.  A  man  having  $2375  in  the  bank  drew  out  $187  at 
one  time  and  $298  at  another;  what  did  he  draw  out  in  all, 
and  what  was  still  remaining  in  the  bank? 

13.  In  1880  there  were  16120  Indians  and  75025  Chinese 
in  California;  there  were  how"  many  of  both,  and  how  many 
more  of  one  than  of  the  other? 

14.  In  a  certain  orchard  containing  425  trees,  187  are 
orange  trees,  153  are  lemon  trees,  and  the  rest  are  nut  trees; 
how  many  nut  trees  are  in  the  orchard? 

15.  Bought  a  horse  for  $185  and  sold  it  for  $212;  how 
much  did  I  gain? 

16.  George  Washington  was  born  in  1732  and  died 
in  1799.  Abraham  Lincoln  was  born  in  1809  and  died 
in  1865.  Which  lived  the  longer,  and  how  many  years 
longer? 


30  CALIFORNIA   SERIES. 

17.  A  farmer  raises  1276  centals  of  wheat;  his  neighbor 
on  the  right  raises  125  centals  more  than  he;  his  left-hand 
neighbor  raises  375  centals  more  than  both  the  others. 
Find  the  number  of  centals  raised  by  each,  and  by  all 
together. 

18.  Daniel  Webster  died  in  1852  at  the  age  of  70;  in 
what  year  was  he  born? 

19.  A  stock-raiser  has  1483  sheep  in  one  corral,  578  in  a 
second,  230  in  a  third,  and  1020  in  a  fourth ;  how  many 
sheep  has  he  ? 

20.  Sold  a  carriage  for  $145,  which  was  $65  less  than  it 
cost  me;  what  did  it  cost  me? 

21.  California  became  a  State  in  1850 ;  how  many  years 
has  it  been  a  State  ? 

22.  From  the  sum  of  309  and  576  subtract  their  differ- 
ence. 

23.  A  speculator  bought  a  lot  of  cattle  for  $2375,  paid 
$450  to  get  them  to  market,  and  sold  them  for  $3100;  how 
much  did  he  gain? 

24.  The  distance  by  rail  from  San  Francisco  to  Ogden  is 
602  miles ;  from  Ogden  to  Oinaha,  1312;  from  Omaha  to 
Chicago,  490  ;  from  Chicago  to  New  York,  963.  Find  the 
distance  by  rail  from  San  Francisco  to  Chicago  ;  from  San 
Francisco  to  New  York. 

25.  Which  is  the  longer  distance  by  rail,  from  San  Fran- 
cisco to  Omaha,  or  from  Omaha  to  New  York,  and  how 
much  longer? 

26.  How  many  years  is  it  since  Columbus  discovered 
America  ? 

27.  The  votes  cast  in  California  at  the  presidential  elec- 
tion of  1884  were  as  follows :  For  Cleveland,  89225 ;  for 
Blaine,  102406  ;  for  St.  John,  2960  ;  for  Butler,  2010 ;  scat- 
tering, 356.     What  was  the  total  vote  of  California? 

28.  Blaine  received  how  many  more  votes  than  Cleve- 
land? 


ARITHMETIC.  81 

^9.  Blaine  received  how  many  more  than  all  the  rest  put 
together? 

30.  I  bought  a  carpet  for  $17,  a  chamber  suit  for  $26,  a 
spring  mattress  for  $8,  a  lounge  for  $18,  an  extension  table 
for  $11,  and  a  parlor  stove  for  $7;  gave  in  payment  $100. 
What  change  should  I  receive? 

31.  The  smaller  of  two  numbers  is  173,  and  their  differ- 
ence is  49;  what  is  the  larger? 

32.  The  sum  of  two  numbers  is  1208,  and  the  larger  is 
749;  what  is  the  smaller? 

33.  The  larger  of  two  numbers  is  970,  and  their  differ- 
ence is  127;  what  is  the  smaller? 

34.  A  man  bought  4  house  lots  for  $4000.  He  paid  $800 
for  the  first,  $125  more  for  the  second  than  for  the  first,  and 
$250  more  for  the  third  than  for  the  second;  what  did  he 
pay  for  the  fourth  ? 

35.  A  boy  said  if  he  had  23  more  marbles  he  Avould  have 
100.     How  many  had  he  ? 

36.  What  number  taken  from  1728  leaves  209? 

37.  Should  a  man  die  to-day  at  the  age  of  69,  in  what 
year  was  he  born? 

38.  If  you  live  till  the  year  1922,  how  old  will  you  be? 

39.  Three  men  go  into  business  together.  The  first  puts 
in  $2500;  the  second,  $1550;  the  third,  $1325.  They  gain 
$725  during  the  year.  How  much  money  have  they  in  all 
at  the  close  of  the  year? 

40.  A  certain  school  has  7  grades.  In  the  first  are  57 
pupils;  in  the  second,.  73;  in  the  third,  61;  in  the  fourth, 
93;  in  the  fifth,  84;  in  the  sixth,  101;  in  the  seventh,  112. 
How  many  pupils  are  in  the  school  ? 

41.  If  273  pupils  in  the  above  school  are  boys,  how  many 
are  girls? 

42.  Benj.  Franklin  was  born  in  1706  and  lived  84  years; 
in  what  year  did  he  die  ? 

43.  The  population  of  the  United   States  in  1870  was 


32  CALIFORNIA   SERIES. 

38567617;  in  1880,  50267519.     How  much  had  it  gained  in 
10  years  ? 

44.  There  were  6608  miles  of  railroad  built  in  1883  in 
the  United  States,  and  11591  miles  in  1882.  How  many 
miles  were  built  in  both  years?  How  many  more  in  1882 
than  in  1883? 

45.  How  many  days  from  Jan.  1  to  July  1? 

46.  Two  boys  have  each  145  cents;  one  gives  the  other 
25  cents.  How  many  cents  has  each  now,  and  how  many 
more  has  one  than  the  other? 

47.  A  man  has  $2783  on  hand  and  owes  $1296 ;  how 
much  is  he  really  worth  ? 

48.  A  man  receives  $125  a  month  for  3  months  ;  he  spends 
during  that  time  $171.     How  much  does  he  save? 

49.  A  man  lays  up  $370  a  year  for  4  years  ;  how  much 
has  he  at  the  end  of  the  time  ? 

50.  A  merchant  bought  3  lots  of  wheat  containing  1250, 
498,  and  726  centals  respectively.  He  sold  550  centals  at 
one  time  and  1500  at  another  ;  how  many  centals  remained  ? 

51.  The  city  of  Rome  was  founded  753  years  before 
Christ  (B.  C).     How  old  is  it? 

52.  The  date  given  for  the  creation  is  4004  B.  C.  The 
Flood  occurred  1652  years  later.  In  what  year  was  the 
Flood? 

53.  Mt.  Everest  is  29062  feet  high  ;  Mt.  Whitney,  14900. 
What  is  the  difference  in  their  heights  ? 

54.  A  lady  went  on  a  journey,  traveling  175  miles  by 
steamer,  213  by  rail,  and  94  by  stage.  What  was  the 
length  of  the  journey? 

55.  A  man  gained  $45  by  selling  a  horse  for  $190.  What 
did  the  horse  cost  him  ? 

56.  How  much  will  a  man  have  left  from  $1000,  if  he 
spends  $125  at  one  time,  $256  at  another,  and  $114  at 
another? 

57.  A  man  sells  130  sheep  for  $325,  115  sheep  for  $345, 


ARITHMETIC.  33 

and  58  sheep  for  $203.     How  many  sheep  did  he  sell,  and 
what  did  he  get  for  all  ? 

58.  A  sells  a  house  to  B  for  $2375;  B  sells  it  to  C  at  a 
gain  of  $250;  C  sells  it  to  D  at  a  loss  of  $175.  What  does 
D  pay  for  the  house? 

59.  A  man  dying  leaves  $3400  to  his  wife,  $1700  to  each 
of  his  two  sons,  and  $1500  apiece  to  his  three  daughters. 
How  much  money  does  he  leave? 

60.  During  the  year  ending  July  1,  1885,  2114  arrests 
were  made  in  Oakland,  of  which  all  hut  906  were  caused 
by  drunkenness.     Find  the  number  thus  caused. 

61.  How  many  days  from  August  1  to  the  end  of  the 
year? 

62.  How  many  years  was  it  from  the  birth  of  Moses  1571 
B.  C.  to  the  founding  of  Rome? 

63.  A  man  exchanged  a  lot  of  wheat  and  $725  for  cattle 
valued  at  $2700.     What  was  the  value  of  the  wheat  ? 

64.  How  many  more  days  are  there  from  June  1  to  Octo- 
ber 1  than  from  Jan.  1  to  May  1? 

65.  The  first  Spanish  mission  founded  in  California  was 
at  San  Diego  in  1769.  79  years  later,  gold  was  discovered 
in  the  State.     In  what  year  was  gold  discovered? 

66.  Mt.  Everest  is  29062  feet  above  the  sea  level;  the 
Dead  Sea  is  1317  feet  below  the  sea  level.  How  many  feet 
does  Mt.  Everest  rise  above  the  Dead  Sea  ? 

67.  A  boy  has  175  cents  but  gives  away  30  to  a  boy  who 
had  none.  After  the  gift,  how  many  more  has  the  first  boy 
than  the  second? 

68.  A  fruit  grower  has  4  rows  of  trees  in  a  certain  orchard, 
containing  32  trees  each.  72  are  orange  trees  and  the 
remainder  are  lemon;  how  many  lemon  trees  are  there  ? 

EXERCISE  48. 

Make  up  10  examples  of  your  own  like  the  preceding, 

work  out,  and  bring  into  the  class  for  dictation  to  the  others. 
3— A 


34  CALIFORNIA   SERIES. 


MULTIPLICATION. 

I  bought  2  apples  for  which  I  paid  2  cents  each ;  what 
did  I  pay  for  both  apples? 

How  do  you  find  it? 

At  2  cents  each,  what  must  I  pay  for  3  apples?  For  4? 
For  5?     ForG?     For  7?     For  8? 

Compare  your  work  with  the  first  direction  in  Example 
1,  Exercise  18.  In  this  work  you  are  adding  by  what  num- 
ber? How  many  times  do  you  take  2  to  get  the  price  of  2 
apples?    To  get  the  price  of  3?     Of  4?     Of  6?     Of  7? 

To  find  how  much  any  number  of  apples,  oranges,  pen- 
cils, etc.,  costs  at  2c.  each,  we  add  by  2's  as  many  times  as 
there  are  apples,  oranges,  pencils,  etc. 

At  3  cents  each  what  will  2  pencils  cost?  3  pencils?  4? 
5?  6?  7?  8?  9? 

Compare  with  the  first  direction  in  Example  2,  Exercise 
18.  You  are  now  adding  by  what  number?  How  many 
times,  for  2  pencils?     For  3?5?7?9? 

Instead  of  adding  from  0  up,  every  time,  when  we  wish 
to  perform  examples  like  the  preceding,  it  is  better  to  com- 
mit to  memory  these  results  for  all  numbers  up  to  10. 

The  process  of  taking  any  number  of  times  a  given 
number  is  called  Multiplication. 

The  number  to  be  taken  a  number  of  times  is  called  the 
multiplicand. 

The  number  showing  how  many  times  the  multiplicand 
is  to  be  taken  is  called  the  multiplier. 

The  result  of  multiplying  is  called  the  product. 

Picl:  out  each  in  the  above  illustrations. 

The  sign  ( X ),  called  times,  is  used  to  indicate  multiplica- 
tion.    Thus, 

3X3=9  is  read  3  times  3  are  9,  or  3  3^s  are  nine. 


ARITHMETIC.  35 

The  multiplicand  and  multiplier  are  sometimes  called 
factors  of  the  product.     Thus,  in  the  phrase, 
3  ^'s  are  6,  3  and  2  are  factors  of  6. 

In  general,  any  whole  numbers,  which,  multiplied  to- 
gether, will  produce  a  given  number,  are  called  factors  of 
that  number. 

EXERCISE  49.    (Oral  and  Written.) 

Add  by  2's  from  0  to  20,  write  out  the  work  in  column  as 
indicated  below,  and  commit  to  memory,  reading  as 
directed  above.     Thus,     1X2=  2 

3X2=  6 

JtX2=  8 

5X2=10 

and  so  on 

to  10X2. 

Do  the  same  with  3's  from  0  to  30;  4's  from  0  to  40;  5's 
from  0  to  50;  6's  from  0  to  60;  7's  from  0  to  70;  8's  from  0 
to  80;  9'sfrom  Oto90. 

EXERCISE  50.    (Oral  and  Written.) 

After  the  thorough  memorizing  of  the  tables,  give  them 
backward,  writing  them  backward,  also. 


EXERCISE  51. 

(Oral.) 

3X2= 

4X3= 

5X6= 

4X9= 

2X9= 

2X3= 

3X4= 

6X5= 

9X4= 

3X6= 

4X2= 

5X3= 

8X5= 

7X7= 

7X9= 

2X4= 

3X5= 

5X8= 

8X7= 

9X7= 

7X2= 

3X7= 

4X7= 

7X8= 

8X9= 

2X7= 

7X3= 

7X4= 

8X8= 

9X8= 

9X2= 

9X3= 

6X7= 

6X6= 

9X9= 

2X9= 

3X9= 

7X6= 

4X4= 

5X9= 

6X2= 

10X3= 

8X6= 

2X8= 

9X5= 

2X6= 

3X10= 

6X8= 

8X2= 

5X5= 

36  CALIFORNIA   SERIES. 

EXERCISE  52.    (Written.) 

Write  all  the  factors  of  the  following  numbers,  and  bring 
in  to  the  class  for  reading.     Thus, 

SI  has  3  and  7  for  its  factors;  hence,  7y^3=^21. 

21,  42,  32,  36,  16,  45,  27,  80,  48,  18,  15,  10,  56,  30,  9,  40, 
25,  64,  14,  28,  56,  20,  60,  63,  81,  70,  24,  72,  90,  8,  12,  54,  11, 
17,  23,  29. 

EXERCISE  53.    (Oral.) 

Use  2  as  a  multiplier  with  each  of  the  following  numbers; 
then  use  3,  4,  5,  6,  7,  8,  9  in  turn: 

7385296410 

A  number  applied  to  a  particular  object  or  thing  is  called 
a  concrete  number;  as,  7  bools,  3  yards,  71  days. 

A  number  used  Avith  no  reference  to  any  object  or  thing 
is  called  an  abstract  number;  as,  7,  3,  71. 

EXERCISE  54.    (Written.) 

Write,  in  one  column,  the  concrete  numbers,  and,  in 
another,  the.  abstract  numbers  in  the  following  : 

51,  29  inches,  7  pencils,  147,  512,  14  cows,  28  horses,  28, 
158,  12  months,  10  cents,  12  knives,  159,  6  dozen  pens, 
1200,  496. 

AVrite  10  abstract  and  10  concrete  numbers  of  your  own. 

When  both  factors  are  abstract,  either  may  be  the  midfi- 
plicand. 

When  one  factor  is  concrete,  it  is  the  multiplicand,  and  the 
product  is  like  it. 

Thus, 

M-r.    M-d.  P-t. 

7X3  units=^21  units;  7X3  tens^Sl  tens;  7Xp=i21. 

At  5  cents  apiece  what  will  7  pencils  cost? 

Model  for  Analysis. — If  1  pencil  costs  5  cents,  7  pencils  will 
cost  7x5  cents,  or*  35  cents. 


ARITHMETIC.  37 

Pick  out  (1)  the  multiplicand  (2)  the  multiplier  (3)  an 
abstract  number  (4)  a  concrete  number.  What  is  the 
product  like  in  name? 

EXERCISE  55.    (Written.) 

-    Write  the  analysis  of  the  following  like  the  preceding 
model: 

1.  At  10  cents  a  dozen  what  will  9  dozen  oranges  cost? 

2.  If  a  watch  ticks  3  times  in  1  second,  how  many  times 
will  it  tick  in  6  seconds  ? 

3.  If  1  yard  contains  3  feet,  how  many  feet  do  8  yards 
contain  ? 

4.  If  1  ton  of  coal  costs  $8  what  will  7  tons  cost? 

5.  What  will  3  pairs  of  shoes  cost  at  $2  a  pair? 

6.  If  a  man  can  walk  4  miles  an  hour,  how  far  can  he 
walk  in  9  hours? 

7.  What  cost  7  cords  of  wood  at  %1  a  cord? 

8.  How  many  trees  are  there  in  an  orchard  containing  9 
rows  of  8  trees  each  ? 

9.  There  are  7  days  in  1  week;  how  many  days  are  in 
^  weeks? 

10.  8  boys  have  6  marbles  each;  how  many  have  all? 

Repeat  the  analysis  orally  in  the  class. 

EXERCISE  56.   (Oral  Analysis.) 

1.  Find  the  cost  of  a  dozen  pencils  at  3  cents  each. 

2.  There  are  4  quarts  in  a  gallon.  How  many  quarts  are 
there  in  7  gallons?  In  4  gallons?  In  9  gallons?  In  5 
gallons  ? 

3.  At  9  cents  a  yard  what  must  be  paid  for  4  yards  of 
calico?     For  7  yards?     For  2  yards?     For  8  yards? 

4.  If  a  man  works  8  hours  a  day,  how  many  hours  does 
he  work  in  5  days?     In  6  days?     In  8  days?     In  3  days? 

5.  I  pay  $5  a  week  for  board.  What  do  I  pay  for  board 
for  2  weeks?     For  4  weeks?     For  6  weeks?     For  7  weeks? 


38  CALIFORNIA   SERIES. 

6.  There  are  10  tens  in  1  hundred.  There  are  how  many 
tens  m  5  hundreds?     In  7  hundreds?     In  9  hundreds? 

7.  A  horse  travels  6  hours  at  the  rate  of  7  miles  an 
hour.     How  far  does  he  travel? 

8.  What  cost  7  2-cent  postage  stamps  ?     5?     8?     4? 

9.  If  6  men  can  dig  a  ditch  in  6  days,  how  long  will  it 
take  1  man? 

10.  At  7  cents  a  yard  what  will  10  yards  of  ribbon  come 
to?     8  yards?     3  yards?     2  yards?     6  yards? 

As  the  multiplicand  may  be  concrete,  representing  hours, 
cents,  etc.,  any  example  in  multiplication  may  be  made  a 
practical  example  like  the  above.     Hence,  instead  of 

6X3=18, 
write 

At  3  cents  each  what  ivill  6  apples  cost? 

EXERCISE  57. 

Form  10  examples  of  your  own,  like  the  above,  from  the 
following,  and  analyze: 

1.9X2=?    3.5X9=?    5.7X7=?    7.    8x5=?      9.3x8=? 
2.6X7=?    4.9X3=?    6.4x4=?    8.10x7=?   10.8x7=? 

Dictate  similar  examples  in  the  class,  on  the  spur  of  the 
moment. 

To  multiply  a  number,  of  several  figures,  by  units. 

Multiply  423  by  2. 

OPERATION. 

4  z  o.  Explanation. — "Write  the  numbers  as  in  Addition. 

2.      Write  the  several  products  in  their  proper  column. 

EXERCISE   58. 

1.  724X2=  4.  123X3=  7.     821x3= 

2.  522X3=  5.  443X2=  8.     610x4= 

3.  321X4=  6.  711X5=  9.  7122x4= 


ARITHMETIC. 


39 


10.  4201X4=  12.  9011X5-= 

11.  2304X2=  13.  7022X4= 

Multiply  538  by  7. 

FULL   OPERATION. 

500+    30+    8=   538 

7  7 


14.  3322X3= 

15.  4232X3= 


Explanation. — 7x8  are  56,  or  5 
tens  6  units.  Write  6  in  units'  place 
and  add  the  5  tens  to  the  tens'  pro- 
duct. 7x3  tens  are  21  tens,  +5  tens 
are  26  tens,  or  2  hundreds  6  tens. 


3500+210+56=3766 
Write  6  tens  in  tens'  column  and  add  the  2  hundreds  to  the  hun- 
dreds' product.  7x5  hundreds  are  35  hundreds, +  2  hundreds  are 
37  hundreds;  which  write  in  hundreds'  column. 

EXERCISE  59.    (Written.) 

Give  oral  explanation  in  the  class: 


1.  5X     25= 

2.  7X  230= 

3.  9X  436= 

4.  4X3198= 

5.  2X4722= 

6.  3X3428= 

7.  4X  409= 

8.  7X4600= 

9.  6X     36= 

10.  5X1008= 

11.  4X  571= 

12.  3X  298= 

13.  9X1019= 


14.  6X   236= 

15.  5X   756= 

16.  8X4008= 

17.  2X   765= 

18.  7X  477= 

19.  8X.  888= 

20.  9X1112= 

21.  6X  746= 

22.  5X4591= 

23.  3X1233= 

24.  7X8769= 

25.  9X9761= 


27.  2X  989= 

28.  4X7017= 

29.  5X5136= 

30.  3X4203= 

31.  6X   547= 

32.  8X7209= 

33.  9X8080= 

34.  7X6700= 

35.  5X2350= 

36.  3X3031= 

37.  4X1105= 

38.  2X7566= 

39.  6X9999= 


26.  4X5005= 
EXERCISE  60.    'Oral  and  Written.) 
]Multiply  the  upper  by  the  lower  number  in  each  exam- 
ple of  Exercise  31. 

EXERCISE  61.    (Oral  AND  Written.) 
Multiply  each  multiplicand  in  examples  1  to  20,  Exercise 

59,  by  every  number  in  turn  from  2  to  9,  except  the  one 

already  given  in  the  example. 

The  same  may  be  done  with  the  other  examples  of  the 

same  exercise. 


40  CALIFORNIA   SERIES. 

To  multiply  by  any  number  of  lO's,  lOO's,  lOOO's,  etc. 

Write  in  figures  and  read  by  common  names  the  follow- 
ing: 

7  10's=  155  lOO's  (hundreds)  = 

15  10's==  176  lOO's  "  = 

25  10's=^  3141  lOO's  "  = 

230  10's=  7  lOOO's  (thousands)  =^ 

175  10's=  12  lOOO's 

3126  10's=  36  lOOO's 

6  lOO's  (hundreds)^  172  lOOO's 

17  lOO's  "  =  230  lOOO's 

20  lOO's  "  =        1756  lOOO's 

But  7  lO's  or  7X10,  is  the  same  as  10  7's  or  10x7;  15 
lO's,  the  same  as  10x15;  175  lO's,  the  same  as  10X175;  6 
100's=:100x6;  3141  100's=100x3141;  12  1000's=1000X 
12;  172  1000's=1000Xl72;  and  so  on. 

Hence,  to  multiply  a  number  by  10,  what  will  you  do? 
By  100?     By  1000?     By  10000?     By  100000? 

7X2  tens=:?     7x2  hundreds^?     7x2  thousands^? 

7X2  lO's  is  the  same  as  2  10'sX7,  or  20x7. 

7X2  lOO's  is  200X7. 

27X200=200X27. 

77X2000=2000X77. 

Hence,  to  multiply  by  2  lO's,  3  lO's,  4  lO's,  etc.,  what  can 
you  do?  By  2  lOO's,  3  lOO's,  4  lOO's,  etc.?  By  2  lOOO's, 
3  lOOO's,  4  lOOO's,  etc.? 

EXERCISE  62.    (Written.) 

Perform  the  multiplications  indicated  below.  Also,  mul- 
tiply by  two  other  numbers  of  lO's,  lOO's,  lOOO's,  besides 
tliose  given.  The  multiplication  by  numbers  with  O's  is 
usually  written  as  below. 

1.  2.  3.  4.  5.  6. 

43     225     75     239     3141     729 
30      70     400     GOO      500     3000 


AlilTIlMJLTliJ. 

7.                       8.                      9. 

10.                  11. 

4280                 2146                 404 

75                 675 

50                    7000               900 

10000             800 

To  multiply  by  numbers  of  two  or 

more  figures. 

Multiply  5847  by  3075. 

FULL  WORK. 

CONTRACTED. 

5847 

5847 

3075 

3075 

29235-=      5X5847 

29235 

409290==     70x5847 

40929 

17541000=3000X5847  17541 


17979525-r3075x5847  17979525 

Since  O's  count  nothing  in  adding,  it  shortens  the  work  to  omit 
writing  them,  as  in  the  contracted  operation. 

Test  by  using  the  multiplicand  foi^  the  midtiplier ;  that  is, 
multiply  3075  by  5847. 

EXERCISE  63. 

Multiply  each  multiplicand,  of  examples  31  to  39,  inclu- 
sive, Exercise  59,  by  each  multiplicand  of  examples  1  to 
10,  inclusive,  and  prove. 

The  same  may  be  done  with  examples  11  to  30  pf  the 

same  exercise. 

EXERCISE  64. 

Write,  perform,  and  prove  10  similar  examples  of  3^our 
own,  and  bring  into  the  class  for  dictation. 

EXERCISE  65.    (Written.) 
See  "  Model  for  Analysis,"  p.  36,  and  analyze  the  follow- 
ing: 

1.  Bought  25  cows  at  $37  a  head;  what  was  paid  for  all? 

2.  There  are  24  hours  in  one  day;  how  many  hours  are 
in  32  days? 

3.  A  lady  paid  $16  a  month  for  board  for  11  months; 
what  was  her  board  bill  ? 


42  CALIFORNIA   SERIES. 

4.  There  are  5280  feet  in  a  mile;  how  many  feet  are  there 
in  18  miles? 

5.  At  an  average  rate  of  23  miles  an  horn*,  how  far  will  a 
railroad  train  go  in  a  day? 

6.  A  man  sets  out  93  acres  of  fruit  trees,  setting  104  trees 
to  the  acre;  find  the  number  of  trees  in  his  orchard. 

7.  What  will  be  the  cost  of  building  35  miles  of  railroad 
at  an  average  expense  of  $33275  a  mile  ? 

8.  Find  the  cost  of  a  ranch  of  960  acres  at  $75  an  acre. 

9.  How  many  pounds  of  tea  are  in  346  chests,  each  chest 
containing  65  pounds  ? 

10.  A  clerk's  salary  is  $75  a  month;  what  does  he  re- 
ceive in  a  year? 

11.  At  $23  each,  what  does  a  furniture  dealer  receive  for 
24  lounges  ? 

12.  A  grocer's  sales  average  $19  a  day  for  the  month  of 
March;  leaving  out  5  days  for  Sundays,  what  were  his 
receipts  during  the  month? 

13.  Each  workman  in  an  iron  foundry  is  paid  $525  a 
year;  what  do  12  men  receive  at  that  rate? 


ARITHMETIC.  4 


o 


DIVISION. 

Copy  the  following  exercise  on  your  slates,  and  in  place 
of  each  hlank  put  how  many  times  the  number  above  the 
line  must  be  taken  to  make  the  one  below: 

786389       10         3  75 


35       72       42        18       48        54        60        27        63        45 

Copy  the  following,  and  place  below  the  line  the  number 
of  times  the  number  on  the  left  of  the  curve  is  contained  in 
the  number  on  the  right: 
5)40   7)56  9)81    8)32    5)50  4)20   10)30  6)54  5)27  8)59 

How  do  the  last  two  differ  from  the  others?  Find  the 
largest  number  below  27  that  is  an  exact  number  of  times 
5.  Find  the  remainder  after  taking  away  5  5's  from  27. 
Do  the  same  with  the  8's  in  59  and  find  what  remains. 

The  process  of  finding  how  many  times  one  number  con- 
tains another,  is  called  Division. 

The  containing  number  is  called  the  dividend. 

The  number  contained  is  called  the  divisor. 

The  number  of  times  the  di\ddend  contains  the  divisor 
is  called  the  quotient. 

The  part  of  the  dividend  left  over,  when  the  divisor  is  not 
contained  an  exact  number  of  times,  is  called  the  remain- 
der.    It  is  always  like  the  dividend. 

Pich  out  each  in  the  ahore  exercise. 

The  sign  (-^)  is  used  to  indicate  division.     Thus, 

35-^7=5, 
Is  read, 

S5  divided  hy  7  is  5.  ' 

It  may  also  be  written  -^  =  5. 


44  CALIFORNIA   SERIES. 

What  precedes  the  sign  (-^.)  in  the  first  expression? 
Where  is  the  same  found  in  the  second  ?  What  follows  the 
sign  (-^)  and  stands  below  the  line? 

EXERCISE   66.    (Written.) 

Write,  in  vertical  column,  the  numbers  from  2  to  20,  by 
2's.  Write  the  sign  (-^)  after  each,  and,  using  2  as  a  divi- 
sor of  each,  find  the  quotient.  After  writing,  practice  upon 
the  exercise.     Thus, 

First  Form,  j  ^"^^""^ 

(  4-^2=2,  and  so  on. 

The  same  by  o's  from  3  to  30  with  3  as  a  divisor,  express- 
ing the  division  in  the  second  form.     Thus, 

( ^=1  -^=3 

Second  Form.  ■]  I     ^         ,  ?     ,         , 

If  =2         V  =  4;  and  soon. 

The  same  by  4's  from  4  to  40,  with  divisor  4,  using  the 
sign  —; 

By  5's  from  5  to  50,  using  divisor  5  and  the  line; 
By  6's  from  6  to  60,  using  divisor  6  and  the  line; 
By  7's  from  7  to  70,  using  divisor  7  and  sign  (-^); 
By  8's  from  8  to  80,  using  divisor  8  and  sign  ( --) ; 
By  9's  from  9  to  90,  using  divisor  9  and  line. 
Bring  into  the  class  to  read. 

To  divide  a  number  is  to  separate  it  into  equal  parts. 

Thus,  when  we  divide  an  apple  into  2  equal  parts,  each 
part  is  one-half;  when  into  3  equal  parts,  each  part  is  one- 
third;  when  into  4  equal  parts,  each  part  is  one-fourth;  and 
so  on. 

So  when  we  divide  a  number  b}^  2,  or  into  2  equal  parts, 
we  get  one-half  the  number;  into  3  equal  parts,  one-third 
the  number;  into  4  equal  parts,  one-fourth  the  number. 
Thus,  dividing  12  by  2  is  taking  one-half  of  it;  or, 
12-^2=one-half  of  12=6,  12---4=:one-fourth  of  12=3, 
12--3=one-third  of  12=4,  12--6=one-sixth  of  12=2, 
and  so  on. 


ARITHMETL 


C 


4 


Ai' 


What  is  each  part  when  we  divide  a  number  into  5  equal 
parts?  Into  6?  7?  8?  9?  10?  11?  12?  15?  17?  19? 
25?     3G?     50?     100?     136?     175? 

Read  the  following  exercise  thus,  One-fifth  of  80  is  6,  etc.: 
5)30    4)28     7)35    6^    9)_54    3)21     2)U    8)J72    10)90 
EXERCISE  67.    (Oral.) 

Use  the  numbers  in  the  left  hand  column  for  divisors  and 
the  other  numbers  in  the  same  row  for  dividends. 

Name  quotients  and  remainders.     Read, 

3  in  15.  5 ;  3  in  29 ^  9  and  2  over ;  and  so  on. 

Also  read, 

■^  (one-third)  of  15  is  5  ;  i  of  29  is  9  and  2  over,  or  P|  (two- 
thirds);  -f  (one-seventh)  of  44  ^^  6  and  2  over,  or  6^  (two- 
sevenths). 


.                   1 

3 

15 

29 

31 

23 

18 

9 

0 

24 

17 

27 

14 

"1 

7 

U 

37 

28 

19 

7 

14 

24 

32 

46 

56 

38 

43 

9 

22 

35 

41 

80 

72 

56 

19 

7 

26 

81 

93 

45  1 

1 

6 

36 

43 

14 

8 

49 

53 

62 

25 

33 

18 

42 

51 

8 

21 

17 

16 

80 

QQ 

44 

55 

0 

38 

14 

35 

76 

5 

35 

43 

52 

24 

37 

15 

8 

28 

49 

36 

21 

18  , 

4 
8 
2 

15 

22 

28 

35 

18 

7 

11 

24 

31 

17 

10 

0 

48 

79 

46 

64 

15 

23 

59 

75 

41 

12 

9 

28 

5 

2 

11 

13 

19 

7 

8 

12 

10 

3 

6 

17 

7 
9 

42 

21 

9 

17 

33 

55 

66 

35 

03 

71 

13 

11 

85 

44 

27 

10 

36 

51 

75 

25 

13 

63 

97 

17 

10 

23 

30 

17 

46 

10 

87 

54 

95 

70 

36 

0 

14 

6 

48 

27 

10 

19 

30 

66 

54 

47 

13 

24 

11 

59 

7 

40 

02 

22 

72 

45 

67 

27 

15 

54 

59 

73 

18 

8 

22 

43 

16 

25 

71 

85 

53 

29 

78 

19 

49 

87 

46  CALIFORNIA   SERIES. 

EXERCISE  68.    (Written  Analysis.) 
If  7  pencils  cost  35  cents,  what  costs  1  pencil? 

Model. — If  7  pencils  cost  35  cents,  1  pencil  costs  }  of  35  cents,  or 
5  cents. 

f  In  this  form  of  analysis  the  dividend  and  quotient 
J       a7^e  concrete  numbers  of  the  same  kind. 
I  The  divisor  is  abstract  and  corresponds  to  the  mul- 
I       tiplier. 

Write  analyses  of  the  following  : 

1.  If  9  dozen  oranges  cost  90  cents,  what  will  1  dozen 
cost? 

2.  If  a  watch  ticks  18  times  in  6  seconds,  how  many 
times  does  it  tick  in  1  second  ? 

3.  How  many  feet  in  1  yard,  if  8  yards  are  24  feet? 

4.  What  is  the  price  per  ton,  when  7  tons  of  coal  cost  $56? 

5.  If  three  pairs  of  shoes  sell  for  $6,  what  do  they  bring 
a  pair? 

6.  How  far  does  a  man  walk  in  1  hour,  if  he  goes  36 
miles  in  9  hours? 

7.  Paid  $49  for  7  cords  of  wood;  how  much  was  the  wood 
a  cord  ? 

8.  9  rows  of  trees  in  an  orchard  contain  72  trees;   how 
many  trees  in  a  row? 

9.  If  there  .are  63  days  in  9  weeks,  how  many  days  in  a 
week? 

10.  I  divided  48  marbles  equally  among  8  boys;  how 
many  did  they  receive  apiece? 

Compare  the  above  work  with  that  of  Exercises  69  and  55. 

EXERCISE  69.    (Written  Analysis.) 

At  5  cents  apiece,  how  many  pencils  can  I  get  for  35 
cents  ? 

Model. — If  1  pencil  costs  5  cents,  I  can  get  as  many  pencils  for 
35  cents  as  5  cents  is  contained  times  in  35  cents,  or  7. 


ARITHMETIC.  47 

[  In  this  form  of  analysis  the  dividend  and  divisor 
are  concrete  numbers  of  the  same  hind. 


Observe  ,   ^,  .         .  ,  ,  , 

The  quotient   is  an   abstract  number  and   corre- 
sponds to  the  nudtiplier  in  Midtiplication. 

Write  analyses  of  the  following  : 

1.  How  many  dozen  oranges  at  10  cents  a  dozen  can  be 
bought  for  90  cents  ? 

2.  How  many  nickel  watches  at  $3  each  can  you  buy  for 
$18? 

3.  There  are  3  feet  in  a  yard;  how  many  yards  in  24  feet? 

4.  At  $8  a  ton,  how  many  tons  of  coal  can  be  bought  for 

$56? 

5.  At  $2  a  pair,  how  many  pairs  of  shoes  can  be  bought 

for  $6? 

6.  At  the  rate  of  4  miles  an  hour,  how  long  will  it  take  a 
man  to  ^valk^G  miles? 

7.  How  many  cords  of  wood  at  $7  a  cord  can  you  buy  for 
$49? 

8.  An  orchard  of  72  trees  has  8  trees  in  a  row;  how 
many  rows  are  there  ? 

9.  In  63  days  how  many  weeks  ? 

10.  I  divided  48  marbles  among  some  boys,  giving  6 
marbles  to  each  boy;  how  many  boys  were  there? 

Compare  work  with  that  of  Exercise  55. 

EXERCISE   70.    (Oral  Analysis.) 

1.  At  3  cents  each,  hoAv  many  pencils  can  vou  buy  for  36 
cents  ?    For  24  cents  ?     For  18  cents  ?    For  30  cents  ? 

2.  There  are  4  quarts  in  1  gallon;  how  many  gallons  will 
28  quarts  make ?     16  quarts?     32  quarts?     24  quarts? 

3.  At  9  cents  a  3^ard,  how  many  yards  of  calico  can  be 
bought  for  36  cents  ?  For  63  cents  ?  For  18  cents  ?  For 
72  cents? 

4.  If  a  man  works  8  hours  a  day,  how  many  days'  work 
will  40  hours  make?    48  hours?     64  hours?     24  hours? 


48  CALIFORNIA   SERIES. 

5.  I  pay  $5  a  week  for  board;  how  many  weeks'  board 
can  I  get  for  $10?     For  $20?     For  $30?     For  $35? 

6.  There  are  10  tens  in  1  hundred;  how  many  hundreds 
will  50  tens  make  ?     70  tens  ?     90  tens  ? 

7.  A  horse  travels  7  miles  an  hour;  in  how  many  hours 
will  he  travel  42  miles  ?     63  miles  ? 

8.  How  many  2-cent  postage  stamps  can  you  get  for  14 
cents?     10  cents?     16  cents?     Scents? 

9.  If  1  man  can  dig  a  certain  ditch  in  36  days,  how 
many  men  will  it  take  to  do  it  in  4  days?  In  9  days?  In 
6  days?     In  12  days? 

10.  At  7  cents  a  yard,  how  many  yards  of  ribbon  can  be 
bought  for  70  cents?  For  21  cents?  For  56  cents?  For 
14  cents?     For  42  cents? 

As  the  divisor  and  dividend  may  represent  cents,  dollars, 
hours,  etc.,  any  example  in  Division  may  become  a  practi- 
cal example.     Thus,  instead  of 

18—3=6, 
Write, 

At  3  cents  each,  hotv  many  apples  can  he  bought  for  18  cents? 

EXERCISE  71.    (Written.) 

Form,  from  the  following,  8  examples  like  the  above, 
and  analyze: 

1.  18--2=?       3.  45-^5=.?       5.  49-f-7=?       7.  40-=-8=:? 

2.  42--6=?       4.  27--9=?       6.  16--4=?      8.  70--10=? 

Give  additional  examples  in  the  class. 

EXERCISE  72.    (Oral  Analysis.) 

(-36  1 
1.  If  9  pencils  cost  -j  on  f  cents,  what  does  1  pencil  cost? 

181  J 
Note. — Use  each  of  the  numbers  in  the  braces  for  one  example. 


ARITHMETIC. 


49 


2.  If  28  quarts  make  7  gallons,  how  many  quarts  are  in 
1  gallon? 

■     '  44  ' 

3.  If  4  yards  of  calico  cost  ^  4c  |^  cents,  what  costs  1  yard  ? 

I      AQ     I 

4.  If  {  r'p  r"  hours  make  8  days'  work,  how  many  hours' 

(34  J    work  to  a  day? 

f  101 

8  ' 

5.  If  I  pay  $  •{  w  r)  r"  for  2  weeks'  board,  what  is  the  rate 


per  week  ? 


12 

I14j 


6.  If  there  are  50  tens  in  5  hundred,  how  many  tens  in 


1  hundred? 


7.  If  a  horse  travels  < 
rate  per  hour? 


00 
42 


54  y  miles  in  6  hours,  what  is  his 
48 
.60J 


8.  I  bought  7  postage  stamps  for  14  cents;  what  did  each 
stamp  cost? 

9.  1  man  can  dig  a  ditch  in  36  days;  how  long  will  it 
take  6  men?  4  men?  Omen?  omen?  12  men?  2  men? 
18  men?    36  men? 

r  801 

10.  If  10  yards  of  calico  cost  ^  90  [>  cents,  what  is  the 
price  of  1  yard?  UOoJ 

f  351 
14  1 

11.  If  7  pen  holders  cast  ^  49  )>  cents,  what  does  1  pen 


holder  cost? 


63 
[56 


12.  If  9  Plymouth  Rock  chickens  cost  $27,  what  is  the 
price  of  1  ? 

EXERCISE  73.    (Written.) 

From  the  examples  given  in  Exercise  70,  form  8  exam- 
ples of  your  own,  similar  to  those  in  Exercise  72. 
4— A 


50 


CALIFORNIA   SERIES. 


SHOET  DIVISION. 

To  divide  numbers,  of  several  figures  each,  by  units. 

Divide  8484  by  4. 


OPERATION. 

4)8484 
2121 


Direction. — Write  the  divisor  at  the  left  of  the  div- 
idend for  convenience  and  begin  with  the  left  figures 
of  the  dividend.  Write  the  quotients  in  their  respect- 
ive columns. 


This  method  of  dividing  has  been  named  Short  Division. 


1.  3699^3= 

2.  3699--9= 

3.  8484--2= 

4.  728--8= 

Divide  43457  by  7. 

FULL   OPERATION. 

7)43457 


EXERCISE  74. 

5.  455--5-= 

6.  217--7= 

7.  1282--2= 

8.  1596--3= 


9.  2505- 

10.  7007- 

11.  1402- 

12.  903- 


42000- 

1400- 

57- 


4345' 


-7= 
-7= 
-7= 


6000 
200 

8  1  over 


6208  1  over 


Explanation. — j  of  43  thousand 
is  6  thousand,  and  1  thousand 
over.  But  1  thousand  is  10  hun- 
dred, which,  with  the  4  hundred 
in  the  dividend,  make  14  hun- 
dred. }  of  14  hundred  is  2  hun- 
dred.  Y  of  57  is  8,  and  1  over.   The 


ciphers  counting  nothing  in  adding,  they  are  omitted  in  the  work, 
as  in  the  contracted  operation.  In  explaining  the  work,  read  thus : 
7  in  43,  6,  and  1  over;  7  in  14,  ^;  7  in  5,  0,  and  5  over;  7  in  57,  8, 
and  1  over,  or  f. 

Test  hy  multiplying  divisor  and  quotient,  and  adding  the 
remainder.     The  residt  should  he  the  dividend. 

EXERCISE  75.    (Written.) 

Give  oral  explanation  in  the  class,  reading  as  in  the  ex- 
planation above,  and  prove  the  work. 

1.  59^2=         4.  88-f-7=        7.  84--4=r         10.     67--3= 

2.  78--3=         5.  99--6=         8.  93-t-5=         11.  796--9= 

3.  97^5=        6.  51--3=:        9.  79-r-2=         12.  576--8= 


ARITHMETIC.  51 

13.  479--6==r  22.  8000--9^  31.  436208-f-3=' 

14.  510-:-5=  23.  8796-f-4=  32.  58436-f-9-= 

15.  708--7=  24.  1001--9=  33.  90000--8= 

16.  429--2=  25.  4296--3=  34.  723506-f-5= 

17.  233-f-4=  26.  6511-^-8=  35.  117452--6= 

18.  420--7=  27.  2458-^2=  36.  89001-^9= 

19.  129---2=  28.  9400-^8.=  37.  76400-^3= 

20.  5309^7=  29.     8057-f-7==  38.     14700-^2= 

21.  7002--5=  30.  23809-^7=  39.  518206h-9= 

EXERCISE  76.    (Written.) 

Divide  each  of  the  dividends  in  examples  31  to  39,  Exer- 
cise 75,  by  each  number  from  2  to  9,  inclusive,  except  the 
one  used  as  divisor  in  the  example.  Finish  with  the  divi- 
dend of  Example  31  by  all  the  divisors,  before  going  to  32. 

This  exercise  may  be  extended  to  as  many  of  examples 
1  to  30  as  the  teacher  mav  desire. 

EXERCISE  77.    (Oral  Review.) 

1.  7+3,--5,X4,— 1,X4,— 3,--5,+7,-^4,x9,+3,--3=? 

2.  15-^3,— l,X4,--8,+7,-^3,X4,--2,--2,x5,—  l,--7=? 

3.  9x4,--6,-l,x8,-f2,--7,X4,--8,-f7,X6,-4,-f-8=? 

4.  17+4,-3 -2,X7,-3,--4,+4,-^3,x9,-f6,+3,--5,-9, 
+0=? 

5.  48--8, X 9,— 4,-10,-3, X  7,+ 1,-5,-3, X 5,-7,+ 1, X 
4,  +  l=? 

6.  9x9,-l,-8,-3,x9,+l,-8,-2,x9,+6,-10,-5,X7, 

-2,+5=? 

7.  3+6,X8— 2,-7,+3,— 5,X10,+10,-10-3,X5,— 2,- 
4,+6,— 10=? 

8.  41  +  7,-6,X2,-l,-3,X4,-2,-2,+l,+4,x3,+5,-7, 
— 5=? 

9.  7xO,+9,x5,-3-G— 4,X8,— 18,X2,— 12,+3,X9,+ 
9,X6=? 

10.  11— 10,+17,— 9,+14— 8,+19,— 3,+20,— 3,+40,— 7, 
+  15,-2-? 


52  CALIFORNIA   SERIES. 

To  divide  by  any  number  of  lO's,  lOO's,  lOOO's,  etc. 

How  many  lO's  in  85,  and  what  remainder?  In  97?  In 
117?     In  376?    In  475? 

If  now  you  move  the  decimal  point  from  the  right  of  the 
nmiiber  one  place  toward  the  left,  you  have,  at  the  left  of 
the  point,  the  quotient  arising  from  dividing  by  10,  and  at 
the  right  of  the  point,  the  remainder.     Thus, 

8.5,  9.7,  37.6, 
Should  be  read, 

8  and  5  over ;  and  so  on. 

How  many  lOO's  in  395,  and  what  over?  In  510?  In 
708?     In  1576?     In  9301  ? 

How  many  places  to  the  left  shall  we  move  the  decimal 
point  to  show  a  division  by  100?     By  1000?     By  10000? 

To  divide  by  20,  since  20  is  2  lO's,  we  divide  first  by  10 
by  moving  the  point  one  place  to  the  left,  and  then  this 
result  by  2.     Thus, 

395—20=39.5—2=19  15  over. 

So  with  30,  40,  50,  and  on  to  90. 

To  divide  by  200,  since  200  is  2  lOO's,  divide  first  by  100 
by  moving  the  point  two  places  to  the  left,  and  this  result 
by  2.     Thus, 

3784-^200=37.8^-^2=18  184  over, 

EXERCISE  78.    (Written.) 

Divide  each  dividend  in  examples  21  to  25,  inclusive. 
Exercise  75,  by  20;  by  30;  by  40;  and  so  on  to  90. 

Divide  each  dividend  in  examples  26  to  30,  inclusive, 
Exercise  75,  by  200;  300;  and  so  on  to  900. 

Divide  each  dividend  in  examples  31  to  35,  inclusive, 
Exercise  75,  by  2000;  3000;  and  so  on  to  9000. 

EXERCISE  79. 

Construct  10  examples  of  your  own  like  the  examples  of 

Exercise  77,  and  bring  into  the  class  for  dictation. 


ARITHMETIC.  53 

LOI^G  DIVISION. 

To  divide  by  numbers  of  two  or  more  figures. 

All  operations  in  division  are  performed  like  Short  Divis- 
ion; but,  in  divisions  by  one  figure,  we  easily  recognize  the 
quotients  and  remainders.  With  large  numbers,  however, 
the  operations  must  be  written,  as  we  cannot  tell  at  sight, 
but  must  find  by  trial,  the  quotients  and  remainders.  This 
process  has  received  the  name  of  Long  Division. 

Divide  1459774  by  239. 

OPERATION.  EXPL-^ATION.-For      COHVeil- 

239)1459774(6107  Quo.     ien^e  write  the  quotient  to  the 

14  34  right  of  the  dividend,  as  we  need 

^  y  the  space  below.    We  find  by  trial 

-  ^  „  multiplication  that  we  can  take  6 

Z^  239's,  or  1434,  out  of  1459  with  25 

187  remainder.    Write  down  the  next 

000  di\adend  figure,  7.    There  is  1  239 

-J  o  rr  A  in  257  with  a  remainder  18;  0  239's 

in  187  with  a  remainder  187;  7 

239's,  or  1673,  in  1874,  with  a  re- 


1^73 
201  Eem.  mainder  201. 


The  following  hints  will  be  found  useful : 

1.  To  find  quotient  figures,  use  the  first  figure  of  the  divi- 
sor and  the  first  one  or  two  figures  of  the  dividend.  Thus, 
in  the  above  example, 

2  in  U;      2  in  2;      2  in  1;       2  in  18. 

2.  One  or  two  dividend  figures  can  be  used  only  in  case 
as  many  more  are  left  in  the  dividend  as  are  left  in  the 
divisor.  Thus,  in  dividend  187,  18  cannot  be  used,  as  it 
leaves  one  figure  only  in  the  dividend,  while  there  are  two 
others  in  the  divisor.  In  such  a  case,  the  figure  in  the  quo- 
tient is  0. 

3.  If  the  second  or  third  divisor  figure  is  large,  take  one 


54  CALIFORNIA   SERIES. 

or  two  less  than  the  number  of  times  the  first  divisor  figure 
is  contained  in  the  first  dividend  figure.     Thus, 

2  in  IJi,  7 ;  but  7  239^s  are  1673,  which  is  too  large. 

4.  Remember :  There  must  always  be  a  quotient  figure 
for  every  figure  brought  down  from  the  dividend. 

EXERCISE  80.    (Written.) 

Divide  each  dividend  in  examples  26  to  35,  Exercise  75, 
by  each  dividend  in  examples  1  to  10.  Prove.  Finish 
with  the  dividend  of  example  26  by  all  the  divisors  before 
going  to  example  27. 

Divide  each  dividend  in  examples  36  to  39,  Exercise  75, 
by  each  dividend  in  examples  11  to  20. 

Divide  each  dividend  in  examples  32,  35,  36,  and  39,  by 
each  dividend  in  examples  26  to  30. 

EXERCISE   81.    (Analysis.) 
Analyze  as  in  Exercise  69 : 

1.  How  many  cows  at  $37  a  head  can  be  bought  for  $925? 

2.  How  many  days  are  there  in  744  hours? 

3.  I  have  $176;  how  many  months'  board  will  it  pay  at 
$16  a  month? 

4.  At  the  rate  of  23  miles  an  hour,  how  many  hours  will 
it  take  a  railroad  train  to  go  552  miles  ? 

5.  How  many  acres  will  9672  fruit  trees  require,  allowing 
104  trees  to  the  acre  ? 

6.  At  $33275  a  mile,  how  many  miles  of  railroad  can  be 
built  for  $1164625? 

7.  Find  the  number  of  acres,  at  $75,  that  can  be  bought 
for  $72000. 

8.  How  many  chests  will  22490  pounds  of  tea  fill,  allow- 
ing 65  pounds  to  a  chest? 

9.  A  clerk  received  $900  salary,  at  $75  a  month;  how 
many  months  did  he  work? 

10.  Find  how  many  .ponies  worth  $55  each  are  in  a  band 
sold  for  $605. 


ARITHMETIC.  55 

GENERAL   PRINCIPLES   OF   DIVISION. 

f  Divideud  4  8  Quotient 
y  Divisor        4 

Multiply  the  dividend  of  A  by  2,  and  divide  that  product 
by  the  divisor  4.  How  does  this  quotient  compare  with  the 
quotient  of  A  ?     How  many  times  greater? 

Multiply  dividend  A  by  3,  and  divide  the  result  by  divi- 
sor A.  Compare  your  quotient  with  quotient  A.  Try  4  for 
a  multiplier  in  the  same  way.     Try  5. 

Fill  out  the  blanks  properly  in  the  following  : 
Multiplying  the  dividend  by  any  number the  quo- 
tient by number. 

Divide  dividend  A  by  2,  and  divide  the  result  by  divisor 
A.  How  does  your  quotient  compare  with  the  quotient  A  ? 
Try  3  for  a  divisor  and  describe  the  result.     Try  4. 

Fill  out : 

Dividing  the  dividend  by  any  number the  quotient 

by number. 

Multiply  divisor  A  by  2,  and  divide  dividend  A  by  the 
product.  Compare  with  quotient  A  and  describe  the  result. 
Try  3  and  4. 

Fill  out : 

^Multiplying  the  divisor  by  any  number  '• the  quotient 

by number. 

Divide  divisor  A  by  2,  and  divide  dividend  A  by  the  re- 
sult.    Compare  with  quotient  A  and  describe.     Try  4. 

Fill  out : 

Dividing  the  divisor  by  any  number the  quotient  by 

number. 

Multiply  both  dividend  A  and  divisor  A  by  2,  and  divide 
the  results.  Multiply  both  by  3;  by  4;  by  5.  Divide  both 
by  2  and  divide  the  results;  by  4. 


56  CALIFORNIA   SERIES. 

Fill  out: 

Multiplying  or  dividing  both  dividend  and  divisor  by  the 
same  number the  quotient. 

PRINCIPLES. 

From  these  illustrations  we  see  that — 

(1)  Multiplying  the  dividend  or  dividing  the  divisor  hy 
any  number,  midtiplies  the  quotient  hy  the  same  number. 

(2)  Dividing  the  dividend  or  midtiplying  the  divisor  by 
any  number,  divides  the  quotient  by  the  same  number. 

(3)  Midtiplying  or  dividing  both  dividend  and  divisor  by 
the  same  number  does  not  change  the  quotient. 

With  the  following,  write  on  your  slate,  operations  similar 
to  those  performed  on  A  : 

1.  60=6  3.  1^=8  5.  ^i'«=10 
10                                     18  10 

2.  |2=6  4.   96=4  6.   ?  =  10 
12                                      24  9 


PRACTICAL  WORK  IN  MULTIPLICATION  AND  DIVISION. 

Examples  in  Multiplication  and  Division  may  be  reduced 
to  one  of  the  following  general  forms  : 

C— What  are  12  times  198? 
D. — How  many  times  is  12  contained  in  144?  or, 
What  is  yV  of  144  ? 


General 

Forms. 


EXERCISE  82. 

Determine  what  is  to  be  found  in  the  following  examples, 
and  write  the  first  20  in  general  form  before  performing. 
Then  analyze. 

1.  A  merchant  sold  50  pieces  of  cloth,  each  containing 
45  yards;  how  many  yards  did  he  sell? 


ARITHMETIC.  57 

2.  Sold  hay  to  the  amount  of  $1728  at  $8  a  ton;  how 
many  tons  did  I  sell? 

3.  What  will  190  acres  of  land  cost  at  $112  an  acre? 

4.  A  mile  is  320  rods;  how  many  rods  are  15  miles? 

5.  Bought  11  chests  of  tea  at  $31  a  chest;  what  did  the 
tea  cost  me  ? 

6.  At  $14  a  ton,  how  many  tons  of  coal  can  be  bought  for 
$1500? 

7.  Bought  15  bales  of  hay,  averaging  235  pounds  to  the 
bale;  how  many  pounds  were  there? 

8.  How  many  bales  of  hay  will  19600  pounds  make, 
allowing  240  pounds  to  a  bale  ? 

9.  A  field  of  600  acres  produced  8700  bags  of  wheat;  how 
many  bags  to  the  acre  was  that  ? 

10.  Sent  to  San  Francisco  5  boxes  of  eggs,  each  contain- 
ing 360  eggs;  how  many  eggs  did  I  send? 

11.  How  many  days  are  in  9785  hours? 

12.  How  many  hours  are  in  365  days? 

13.  Sold  850  head  of  cattle  at  $28  a  head;  how  much 
money  did  I  receive? 

14.  How  many  calves  at  $14  can  I  purchase  for  $1974? 

15.  I  Avalked  3  days,  8  hours  per  day,  and  found  I  had 
gone  72  miles;  what  was  my  average  rate  per  hour?  ^ 

16.  A  man  saves  $175  a  year  for  11  years;  how  nuich 
does  he  save  in  the  time  ? 

17.  In  one  year  in  the  United  States  132890  tons  of  lead 
were  produced,  worth  $95  a  ton;  find  the  whole  value. 

18.  137  mills  in  California  in  1882  produced  1246453 
sacks  of  flour;  what  was  the  average  per  mill? 

19.  The  State  raised  3672  centals  of  buckwheat  from  297 
acres;  find  the  average  per  acre. 

20.  At  25  cents  a  day  what  will  a  man's  cigars  cost  him 
in  1  year? 

21.  Paid  $512  for  64  tons  of  hay;  what  will  25  tons  cost 
at  the  same  rate  ? 


58  CALIFORNIA   SERIES. 

22.  If  a  family  spend  15  cents  a  day  for  beer,  how  much 
is  spent  in  4  weeks?  How  many  loaves  of  bread  at  10 
cents  each  would  the  money  buy  ? 

23.  Bought  15  cows  at  $25  a  head,  11  horses  at  $95,  and 
50  sheep  at  $3 ;  what  did  the  whole  cost  me  ? 

24.  Bought  15  acres  of  land  of  one  man  for  $1575,  and 
25  acres  of  another  for  $2750;  which  cost  me  the  more  per 
acre,  and  how  much? 

25.  I  have  $2000.  I  buy  a  lot  of  land  for  $295;  build 
a  house  on  it  for  $1275,  a  shed  for  $96,  and  set  out  trees 
which  cost  me  $12;  buy  a  horse  for  $115,  and  5  tons  of  hay 
at  $12  a  ton.  With  the  remainder  of  my  money  I  buy  3 
acres  of  pasturage;  what  do  I  pay  per  acre? 

26.  Exchanged  8  rolls  of  butter  for  2  pounds  of  tea  at 
65  cents,  1  pound  of  coffee  at  35  cents,  10  pounds  of  sugar 
at  8  cents,  25  pounds  of  flour  at  3  cents,  and  2  pounds  of 
honey  at  20  cents;  what  did  I  receive  a  roll  for  my  butter? 

27.  Bought  15  sacks  of  potatoes  for  1275  cents,  and  sold 
them  for  10  cents  a  sack  more  than  I  paid  for  them;  what 
did  I  get  for  them,  and  how  much  more  than  I  paid  ? 

28.  Bought  95  centals  of  wheat  at  $1  a  cental;  if  I  give  5 
20-dollar  pieces  in  payment,  what  change  do  I  receive? 

29.  What  must  I  pay  for  10  pounds  of  oatmeal  at  5  cents, 
4  rolls  of  butter  at  75  cents,  and  2  dozen  eggs  at  25  cents  ? 

30.  After  taking  37  oranges  from  a  box,  there  were  13 
more  than  twice  as  many  left  in  the  box;  how  many  oranges 
were  in  the  box  before  any  were  taken  out? 

31.  At  $21  a  barrel  how  many  barrels  of  sugar  can  be 
bought  for  $3675? 

32.  Suppose  each  barrel  in  the  31st  example  contained 
265  pounds;  what  was  the  total  weight? 

33.  A  ship  sails  4032  miles  in  14  days;  how  many  miles 
a  day  does  she  sail?     How  many  miles  an  hour? 

34.  Sold  35  bales  of  cloth,  each  bale  containing  41  yards; 
how  many  yards  did  I  sell  ? 


ARITHMETIC.  59 

35.  A  cattle  dealer  bought  175  head  of  cattle  at  $24  a 
head,  paying  $3500  cash  down;  how  much  remained  to  be 
paid? 

36.  $3  a  day  amounts  to  how  much  in  4  weeks? 

37.  Subtract  3  thousand  8  hundred  79  from  4  thousand, 
multiply  the  remainder  by  1  hundred  21,  add  17  hundred 
81  to  the  product,  and  divide  the  sum  by  23;  what  is  the 
quotient  ? 

38.  Two  men  start  from  the  same  place  and  travel  in  the 
same  direction,  one  at  the  rate  of  23  miles  a  day,  the  other, 
28  miles;  how  far  apart  are  they  at  the  end  of  13  days? 
Draw  a  diagram  on  j^our  slate  to  show  this. 

39.  Two  men  start  from  places  600  miles  apart  and  walk 
towards  each  other;  one  travels  20  miles  a  day,  the  other  29 
miles;  how  far  apart  are  they  at  the  end  of  11  days?  Draw 
diagram. 

40.  The  first  census  of  the  United  States  was  taken  in 
1790;  since  then  a  census  has  been  taken  every  10  years; 
how  many  had  been  taken,  up  to  1887? 

41.  The  first  president  of  the  United  States  was  inau- 
gurated in  1789;  since  then  a  president  has  been  inaugu- 
rated every  4  years;  how  many  inaugurations  had  there 
been,  up  to  1887? 

42.  In  California,  in  the  year  ending  June  30,  1880,  5 
factories  produced  $159175  worth  of  silk  goods  ;  what  was 
the  average  per  factory? 

43.  If  a  man  spends  20  cents  a  day  for  whisky,  and  25 
cents  for  cigars,  what  will  he  spend  in  4  years?  At  50 
cents  a  day,  how  many  days'  board  would  the  money  fur- 
nish to  a  disabled  soldier  ? 

44.  How  many  bales  of  cotton  in  259186  pounds,  allow- 
ing 312  pounds  to  a  bale? 

45.  Which  goes  farther,  a  railroad  train  in  6  days  at  the 
rate  of  22  miles  an  hour,  or  a  steamship  in  7  days  at  the 
rate  of  16  miles  an  hour?     How  much? 


60  CALIFORNIA   SERIES. 

46.  Find  the  cost  of  137  cords  of  wood  at  $13  a  cord. 

47.  I  have  two  fields  of  160  acres  each.  From  one  I  cut 
2  tons  of  hay  to  the  acre  and  sell  it  for  $11  a  ton;  from  the 
other  I  get  16  centals  of  wheat  to  the  acre  and  sell  it  at 
$1  a  cental;  tind  what  I  received  for  both,  and  how  much 
more  for  one  than  for  the  other. 

48.  I  feed  my  cow  32  pounds  of  hay  a  day;  how  long  will 
8  bales,  of  240  pounds  each,  last? 

49.  How  many  sacks  of  flour  will  2750  j^ounds  make,  at 
50  pounds  to  a  sack  ? 

50.  Bought  10  horses  at  $125  each  and  paid  for  them  in 
wood  at  $10  a  cord;  how  many  cords  did  it  take? 

51.  A  fruit-grower  sets  out  11984  trees  on  107  acres  of 
land;  how  many  trees  to  the  acre  ? 

52.  If  12  men  can  dig  a  ditch  in  12  days,  how  long  will 
it  take  1  man? 

53.  How  many  boxes  of  oranges  at  $1  a  box  can  I  buy 
for  $375? 

54.  At  an  average  of  750  oranges  to  a  tree,  how  many 
oranges  are  in  an  orchard  of  84  trees?     How  many  dozen? 

55.  What  are  they  worth  at  12  cents  a  dozen? 

56.  If  12  men  can  build  a  house  in  16  days,  how  long 
will  it  take  6  men  ? 

57.  Bought  150  barrels  of  flour  for  $750;  sold  125  barrels 
at  $6  a  barrel,  and  the  remainder  at  $4;  how  much  did  I 
gain  ? 

58.  How  many  65-dollar  gold  watches  can  you  buy  for 
$1000,  and  how  many  5-dollar  gold  rings  for  the  remainder? 

59.  A  man  receives  $120  a  month;  his  expenses  are  $60 
a  month ;  how  long  will  it  take  him  to  pay  for  a  house  that 
cost $1920? 

60.  A  man's  salary  is  $1500  a  year;  he  pays  $22  a  month 
for  board,  and  $42  a  month  for  additional  expenses;  what 
will  he  save  in  4  years  ? 

61.  A  man  bought  a  horse  for  $175;   he  kept  him  24 


ARITHMETIC.  61 

weeks  at  an  expense  of  -^2  a  week  and  then  sold  him  for 
$225;  what  did  he  gain? 

62.  If  a  man  deposits  $15  in  the  bank  every  month  from 
the  time  he  is  21  years  old  until  he  is  70,  how  much  will  he 
then  have  deposited  ? 

63.  The  President  of  the  United  States  receives  $50000  a 
year;  what  does  he  get  a  month?     In  1  term  of  office? 

64.  I  pay  $1974  for  141  head  of  calves;  how  much  do  I 
pay  per  head  ? 

65.  A  man  bought  150  calves  at  $14  a  head  and  sold 
them  so  as  to  gain  $450;  what  did  he  get  a  head? 

66.  What  is  the  average  value  of  5  horses,  worth  respect- 
ively $85,  $95,  $105,  $115,  and  $120? 

67.  A  man  receives  a  salary  of  $1750;  his  expenses  are 
$3  a  day;  how  much  does  he  save  in  a  year? 

68.  If  23  acres  of  land  cost  $1955,  what  will  33  acres  cost 
at  the  same  rate  ? 

69.  Bought  a  certain  number  of  watches  for  $432;  sold 
them  for  $20  apiece,  gaining  $2  on  the  cost;  how  many 
watches  were  there? 

70.  A  farmer  had  784  sheep;  he  sold  200  at  one  time  and 
375  at  another;  what  are  the  remainder  worth  at  $2  a  head? 

71.  I  have  3  fields  containing  320  acres  in  all  and  worth 
$30000;  the  first  contains  160  acres  worth  $125  an  acre,  the 
second  80  acres  worth  $75  an  acre;  what  is  the  value  per 
acre  of  the  third  ? 

72.  Bought  31  hogs  at  $3  each,  and  gave  in  payment 
eight  10-dollar  bills  and  three  5-dollar  gold  pieces;  what 
change  should  I  receive? 

73.  A  liquor  dealer  bought  15  casks  of  brandy,  each  con- 
taining 38  gallons,  at  $4  a  gallon;  find  the  cost. 

74.  Find  the  average  value  of  4  lots  of  land  worth  re- 
spectively $195,  $210,  $255,  and  $300. 

75.  In  1  gallon  there  are  231  cubic  inches;  how  many 
cubic  inches  in  63  gallons? 


62  CALIFORNIA   SERIES. 

76.  A  miller  has  11  tons  of  flour  valued  at  $45  per  ton; 
he  adds  to  it  4  tons  at  the  same  value,  and  then  sells  8  tons; 
what  is  the  whole  value  of  the  part  left  ? 

77.  I  paid  $2160  for  16  horses;  4  of  them  being  stolen, 
for  what  must  I  sell  the  rest  apiece  that  nothing  may  be 
lost? 

78.  If  35  yards  of  cloth  cost  $105,  what  cost  25  yards  ? 

79.  A  merchant  sold  two  pieces  of  cloth  for  $296;  one 
piece  contained  32  yards,  the  other,  42  yards;  what  aver- 
age price  per  yard  did  he  receive  ? 

80.  James  has  94  marbles,  which  are  2  less  than  4  times 
as  many  as  John  has;  how  many  has  John? 

81.  Mary  washes  dishes  for  her  mother  15  minutes  every 
morning;  if  she  receives  10  cents  for  an  hour's  work,  how 
much  money  will  she  earn  in  4  weeks? 

82.  John  wished  to  know  his  father's  and  mother's  ages; 
his  father  told  him  the  product  of  their  ages  was  1755,  and 
his  mother's  age  was  39;  how  old  was  his  father? 

83.  How  many  pounds  of  cheese  at  15  cents  a  pound  are 
worth  135  gallons  of  milk  at  25  cents  a  gallon? 

84.  A  certain  school  has  4  rooms,  with  an  average  of  65 
scholars  to  a  room;  if  105  scholars  are  boys,  how  many 
girls  are  in  the  school  ? 

85.  A  field  has  two  of  its  sides  105  rods  each,  and  the 
other  two  108  rods  each;  how  long  is  the  fence  surrounding 
the  field?     Draw  a  diagram  of  the  field. 

86.  A  tower  is  148  feet  high;  hoAV  many  steps,  each  6 
inches  high,  will  it  take  to  reach  the  top? 

87.  A  certain  quantity  of  barley  lasts  11  horses  15  days; 
how  long  would  it  last  5  horses  ? 

88.  Bought  9  horses  for  $1530,  and  sold  them  for  $1665; 
how  much  did  I  gain  on  each  horse  ? 

Make  up  10  examples  of  your  own  like  the  above,  work, 
and  bring  to  the  class  for  dictation. 


ARITHMETIC.  6 


Q 


FACTORS. 

What  are  Factors?     (See  p.  35.) 

Review  Exercise  52.  What  is  peculiar  in  the  last  4  num- 
bers of  that  exercise  ? 

An  integral  exact  divisor  of  a  number  is  a  factor  of  it. 
A  number  that  contains  factors  is  a  composite  number. 
A  number  that  contains  no  factors  is  a  prime  number. 
Factors  which  are  themselves  prime  numbers  are  prime 
factors. 

Pick  out  illustrations  of  each  in  Exercise  52. 

To  find  the  prime  factors  of  a  number. 

Find  the  prime  factors  of  60. 

OPERATION. 

Explanation. — Divide  the  given  number  by  its  small- 
est prime  factor;    the  quotient  by  its   smallest  prime 
factor;    and  so  on  until  the  quotient  is  prime.      The 
quotient  and  tlie  several  divisors  are  the  prime  factors, 
o 

Test  by  talcing  the  product  of  the  prime  factors,  which 
should  be  the  number  itself. 

When  the  same  number  occurs  in  another  several  times 
as  a  factor,  the  number  of  times  it  occurs  is  shown  by  a 
small  figure  placed  to  the  right  and  above  it,  and  called 
the  power.  Thus,  2,  as  a  factor,  occurs  twice  in  60.  Write 
it — 2'-]  read  it — 2  second  power. 

The  following  hints  will  be  found  useful  in  factoring. 

A  number  is  divisible: 

By  2  or  5,  if  its  units  figure  is  divisible  by  2  or  5,  respect- 
ively; 

By  3,  if  the  sum  of  its  figures  is  divisible  by  3. 

All  higher  prime  factors  than  these  are  usually  found  by 
trial  division.     Try  the  prime  numbers  as  divisors  in  their 


2 

60 

2 

30 

o 
o 

15 

64 


CALIFORNIA   SERIES. 


order  upward,  commencing  with  2,  until  you  reach  one 
whose  quotient  is  no  larger  than  itself.  If  none  of  these 
are  contained,  the  number  to  be  factored  is  prime. 

Composite  numbers  need  not  be  used  as  divisors,  since 
every  composite  number  is  made  up  of  some  smaller  prime 
numbers  than  itself,  which  prime  numbers  you  have  al- 
ready tried;  and  no  number  contains  a  composite  number 
as  a  factor,  unless  it  contains  all  the  prime  factors  of  that 
composite  number.  Thus,  a  number  divisible  by  6  is  also 
divisible  by  2  and  3,  the  factors  of  6. 

EXERCISE  83. 

Make  a  list  of  prime  numbers  to  100,  as  follows: 

1.  Write  all  the  numbers  in  order,  from  2  to  100; 

2.  Since  every  second  number  after  2  is  divisible  by  2, 
cross  it  out; 

3.  Cross  out  every  third  number  after  3,  because  divisi- 
ble by  3; 

4.  Every  fifth  number  after  5,  because  divisible  by  5: 

5.  Lastly,  every  seventh  number  after  7,  because  divisi- 
ble by  7. 

Those  not  crossed  are  the  prime  numbers  sought. 


EXERCISE  84.    (Oral.) 


Determine  by  inspection  which  of  the  following  numbers 
are  divisible  by  2,  3,  or  5: 


1. 
2. 

3. 

4. 


f  1351 
1  270  I 
\  207  ^ 
I  1017 


7021 
4706  [j 
f  3003  (  " 
\  11011 


5  f  ^25 
^'  \  4350  ^ 

^'  \    203  J 

«  f  597  (  ^ 
^'  \  237  j 


9. 
10. 

11. 
12. 


9751 
555 
510 
714 


>e 


3781 
1818 
,  1011  , 
I  1101  J 


kf 


13. 
14. 


17.  279,  496. 


15. 
16. 
18.  213,  284. 


(246] 
1438  I 
f720f^ 
J256j 

9811 

846  ^^ 
329  J 


ARITHMETIC.  65 

EXERCISE  85.    (Written.) 

Find  the  prime  factors  of  all  the  numbers  of  Exercise  86; 
same  in  Exercise  84. 


GEEATEST   COMMON  FACTOE. 

Pick  out  a  factor  that  is  found  in  all  the  numbers  in  each 
of  the  following  sets: 

1.  16  and  20.  3.  12  and  16.  5.  6,  12,  and  24. 

2.  25  and  20.  4.  4,  8,  and  12.  6.  9,  12,  and  18. 

Pick  out  the  largest  factor  found  in  all  the  numbers  of 
each  of  the  preceding  sets. 

A  factor  contained  in  each  of  several  numbers  is  called 
a  common  factor  of  those  numbers. 

The  greatest  factor  common  to  several  numbers  is  called 
their  greatest  common  factor  (g.  c.  f.). 

Name  each  in  the  above  sets. 

Numbers  that  have  no  common  factors  are  prime  to  each 
other. 

To  find  the  g.  c.  f.  of  several  numbers. 

What  is  the  greatest  factor  common  to  30,  45,  and  75? 

OPERATION.  Explanation. — Separate  the  simplest 

30=2 X  3X  5  number  into  its  prime  factors.    Of  these, 

45  reject  such  as  are  not  contained  in  all 

75     3x5=15=g.  C.  f.      the  other  numbers.    The  product  of  the 

remaining  factors  is  the  g.  c.  f.     Or, 
3  )  30 — 4  5 — 7  5         Divide  by  as  many  common  prime  factors  as 
5  VTo     15     25^     ^^^^  ^®  found;  their  product  is  the  g.  c.  f. 

2       3        5 

3X5=15  g.  c.  f. 
5— A 


66  CALIFORNIA   SERIES. 

EXERCISE    86.    (WRITTEN.) 

Find  the  g.  c.  f.  of: 

1.  24,  36,  42.  13.  105,  140,  175.       25.  55,  110. 

2.  33,  44,  77,  187.    14.  99,  180,  252.         26.  81,  120,  141. 

3.  120,  144,  216.      15.  132,  154,  165.       27.  78,  169,  130. 

4.  135,  180,  90.        16.  60,  80,  100,  120.  28.  150,  210,  330. 

5.  108,  45,  81.  17.  864,  420,  600.       29.  99,  132. 

6.  85,  95.  18.  75,  105,  120.         30.  120,  165. 

7.  72,  168.  19.  108,  252.  31.  120,  252. 

8.  119,  132.  20.  39,  52,  65.  32.  85,  102. 

9.  24,  33,  120.  21.  84,  132.  33.  42,  77,91. 

10.  36,  44,  144.  22.  168,  539.  34.  34,  44; 

11.  105,  120,  135.      23.  112,  147,  168.       35.  28,  98. 

12.  144,  180.  24.  287,  369.  36.  110,  210. 

EXERCISE  87.    (Oral.) 
Name  the  g.  c.  f.  at  sight: 


1. 

4,  6,  8. 

11. 

15,  21. 

21. 

56,  77. 

2. 

9,  12,  15. 

12. 

21,  28. 

22. 

27,  45. 

3. 

10,  15,  20. 

13. 

16,  20. 

23. 

35,  45,  48. 

4. 

4,  8,  12. 

14. 

40,  45. 

24. 

32,  40,  56. 

5. 

7,  14,  21. 

15. 

36,  45. 

25. 

72,  32. 

6. 

9,  18,  27. 

16. 

36,  54. 

26. 

18,  32,  40. 

7. 

10,  20,  30. 

17. 

16,  32. 

27. 

25,  50. 

8. 

16,  24,  32. 

18. 

63,  81. 

28. 

30,  60. 

9. 

3,  4,  5. 

19. 

56,  64. 

29. 

12,  32,  44. 

10. 

18,  30,  36. 

20. 

EXERCISE 

33,  44. 

88.    (Written.) 

30. 

24,  27,  33. 

Find  the  g.  c.  f.  of  the  sets  of  numbers  included  under 
the  same  figure,  Exercise  84. 

Also  of  each  set  of  numbers  included  under  the  same 
letter,  Exercise  84. 

Sometimes  the  prime  factors  are  not  easily  recognized, 
nor  found  except  by  long  trial.     In  such  case  the  work  may 


ARITHMETIC.  67 

be  shortened  by  dividing  the  larger  number  by  the  smaller, 
and  factoring  the  remainder;  ascertaining  whether  any  of 
its  factors  are  common  to  the  smaller  nmnber. 

Thus; 

Find  the  g.  c.  f.  of  629  and  731. 


OPERATION.  Explanation. — Multiplication  is  the  suc- 

62  9)731(1  cessive  additions  of  the  same  number  a 

(^29  certain  number  of  times.     If  two  numbers 

"T-T-r 9wO\/-i7    have  a  common  factor,  each  may  be  ob- 

tallied  by  successive  additions  of  that  fac- 

1  7)629  tor.     Here  629  is  obtained  by  adding  37 

37  17's.     Any  number  above  629  containing 

the  factor  17  is  obtained  by  adding  a  sufii- 

.  • .   1  /=  g.  c.  f.  ^.^^^^  number  of  17's  to  629.    But  the  sum 

37X1  7^^6  2  9  of  the  17's  added  to  629  to  produce  731  is 

ay'-\  7__  102         ^^®  same  as  the  difference  between 629 and 


43X17=731 
factor  of  their  difference. 


731.     That  is, 
A  common  factor  of  two  numbers  is  also  a 


EXERCISE    89.    (Written.) 

Find  by  the  previous  method  the  g.  c.  f.  of:" 

1.  168,  539.  3.  287,  369.  5.  371,  636. 

2.  147,  168.  4.     78,169.  6.  279,^961. 

Also,  sets  2,  4,  6,  11,  15,  and  16,  Exercise  84. 


MULTIPLES. 


Name  2  numbers  that  have  both  4  and  3  as  factors;  7 
and  2;  9,  6,  and  3;  8,  4,  and  6;  5  and  10. 

Name  the  smallest  number  in  each  case  that  contains 
the  given  factors  above. 

A  number  which  contains  another  as  a  factor  is  called  a 
multiple  of  that  factor. 


68  CALIFORNIA   SERIES. 

A  number  which  contains  several  others  as  factors  is  a 
common  multiple  of  those  factors;  if  the  smallest  number, 
the  least  common  multiple  (1.  c.  m.). 

Name  an  example  of  each  in  the  above  sets. 

To  find  the  1.  c.  m.  of  several  numbers. 


20 
15 


Find  the  1.  c.  m.  of  9,  12,  15,  and  20. 

OPERATION. 


2 

9     12     15 

-20 

2 

9—  6—15- 

-10 

3 

9—  3—15- 

-   5 

5 

3               5 

5 

Explanation. — Multiply  the  largest  num- 

0  I  20x3x3=180    ^®^  ^^  ^^^  ^^^  prime  factors  found  in  the 
"^  I  other  numbers,  but  not  contained  in  this. 

12J 

Divide  by  any  prime  factor  common  to 
two  or  more,  and  the  quotients  in  the  same 
way  until  prime  to  each  other.  The  prod- 
uct of  the  divisors  and  final  quotients  is 
the  1.  c.  m. 
3 

EXERCISE  90.    (Written.) 
Find  the  1.  c.  m.  of: 

1.  30,  45,  90.  14.  12,  16,  20,  24.  27.  18,  27,  36. 

2.  24,  36,  42.  15.  28,  42,  35.  28.  26,  39,  65. 

3.  4,  8,  10,  5.  16.  50,  75,  125.  29.  33,  44,  55. 

4.  5,  12,  15,  30.  17.  9,  10,  12.  30.  84,  96. 

5.  7,  12,  18,  24.  18.  24,  30,  36,  40.  31.  12,  13. 

6.  75,  100.  19.  108,  132,  144.  32.  13,  16. 

7.  20,  30,  40.  20.  7,  11,  14,  21.  33.  23,  25,  30. 

8.  33,  44,  21.  21.   72,  84,  132.  34.  9,  11. 

9.  105,  120.  22.  75,  105,  120.  35.  24,  26. 

10.  18,  27,  12.  23.  30,  42,  126.  36.  24,  25. 

11.  14,  21,  15.  24.  120,  140,  210.  37.  64,  84. 

12.  3,  4,  5,  6,  10.  25.  15,  21,  35.  38.  34,  36. 

13.  12,  18,  24,  36,  72.  26.  38,  57,  95.  39.  17,  18,  30. 

EXERCISE  91.    (Oral.) 
Name  at  sight  the  1.  c.  m.  of: 
1.  2,  3,  4,  6.  2.  2,  11.         3.  2,  3,  4,  6,  8,  12,  16. 


ARITHMETIC.  69 

4.  6,  9,  18.  13.  6,  8,  12,  24.  22.  5,  10,  25,  50. 

5.  4,  5,  10.  14.  5,  6,  10,  15.  23.  2,  3,  4,  6,  9,  12, 18. 

6.  3,  5.  15.  5,  7.  24.  2,  3,  4,  5,  6,  10,  12, 15. 

7.  5,  8,  10.  16.  4,  9,  6.  25.  2,  3,  6,  7,  14. 

8.  7,  9.  17.  5,  10,  20.  26.  3,  5,  9,  15. 

9.  6,  8,  9,  12.  18.  4,  5,  6.  27.  2,  3,  6,  9,  18. 

10.  5,  10,  15.      19.  5,  10,  12.         28.  3,  6,  9,  12,  36. 

11.  3,  5,  15.        20.  5,  11.  29.  2,  3,  4,  6,  8,  9,  12. 

12.  4,  8,  16.        21.  3,  7.  30.  3,  7,  9. 

When,  in  finding  the  1.  c.  m.  of  two  numbers,  the  numbers 
are  not  easily  factored,  it  is  well  to  find  the  g.  c.  f.  by  the 
division  method ,  divide  one  of  the  numbers  by  it,  and  mul- 
tiply the  quotient  by  the  other.     Thus, 

Find  the  1.  c.  m.  of  119  and  187. 

OPERATION. 

119)187(1 
119 
68=2X2X17 

17)119(7        7X187=1309  1.  cm. 

EXERCISE  92.    (Written.) 
Find,  by  the  preceding  method,  the  1.  c.  m.  of: 

1.  105,  189.         3.     78,  169.        5.  168,  539.        7.  147,  168. 

2.  91,  169.         4.  119,  132.        6.  287,  369.        8.  279,  124. 

Also,  examples  11  to  20,  Exercise  87,  and  numbers 
marked  2,  4,  6,  7,  11,  15,  and  16,  Exercise  84. 

EXERCISE    93. 

AVrite  and  perform  10  examples  of  your  own  in  g.  c.  f. 
and  10  in  1.  c.  m.,  and  bring  to  the  class  for  dictation. 


70  CALIFORNIA   SERIES. 


PRACTICAL  WORK  IN  FACTORING. 


f  E. — Find  the  largest  factor  common  (g.  c.  f.)  to 
General  !  48  and  60. 

Forms.    |  F. — Find  the  smallest  number  that  will  contain 

1^  (1.  c.  m.  of)  12  and  16. 


EXERCISE  94. 

Write  out  the  following  examples  in  proper  general  form 
before  performing : 

1.  I  have  two  rooms  respectively  15  and  18  feet  wide; 
what  is  the  widest  carpeting  that  will  exactly  fit  the  rooms? 

2.  A  man  wishes  to  fence  a  field  having  three  sides  re- 
spectively 120,  128,  and  144  feet  long;  what  is  the  length  of 
the  longest  rail  he  can  use,  and  not  cut  the  rails? 

3.  Three  men  can  walk  3,  4,  and  5  miles  an  hour  respect- 
ively; what  is  the  length  of  the  shortest  journey  they  can 
walk,  and  each  walk  an  exact  number  of  hours? 

4.  A  man  has  two  lots  of  land  360  and  480  feet  wide 
respectively,  which  he  wishes  to  divide  into. house-lots  of 
equal  widths;  what  is  the  greatest  width  of  house-lot  he 
can  make,  and  how  many  will  there  be? 

5.  Two  boys  travel  around  a  race-track  80  rods  in  circum- 
ference, starting  together,  one  making  the  circuit  in  15  min- 
utes, the  other  in  20;  in  what  time  will  they  be  together 
again  at  the  point  of  starting?  How  many  rods  will  each 
have  traveled? 

6.  What  is  the  smallest  tank  that  can  be  filled  by  using 
a  4-quart,  6-quart,  8-quart,  or  10-quart  measure? 

7.  A  man  sends  to  market  525  pounds  of  barley  and  945 
pounds  of  wheat  in  the  largest  sized  bags  he  can  use  and 
have  each  contain  the  same  number  of  pounds:  how  many 
pounds  were  there  to  a  bag  ?  What  was  each  kind  of  grain 
worth  at  $2  a  bag? 


AFdTHMETIC.  71 

8.  I  wish. to  spend  the  smallest  like  sum  for  pencils  at 
5,  6,  7,  8,  and  10  cents  each;  find  it. 

9.  A  man  spends  equal  sums  in  buying  2-cent,  3-cent, 
5-cent,  and  10-cent  postage  stamps,  using  the  smallest  sum 
possible;  what  did  all  the  stamps  come  to? 

10.  Place  112  oranges  and  140  lemons  in  piles,  without 
mixing,  so  that  each  pile  shall  have  the  same  number,  and 
that  the  largest  possible. 

11.  Draw  from  a  basket  of  nuts  3  lots  of  equal  numbers, 
the  smallest  possible,  so  as  to  arrange  the  3  lots  in  piles  of 
7,  9,  and  12  nuts,  respectively. 

12.  A  boy  has  the  same  number  of  marbles  in  each  of  4 
boxes;  the  first  he  arranges  in  piles  of  3  each,  the  second  in 
piles  of  4  each,  the  third  in  piles  of  5  each,  and  the  fourth 
in  piles  of  6  each;  on  calculation  he  found  he  had  the  few- 
est marbles  to  a  box  that  could  be  so  arranged;  how  many 
marbles  had  he? 

13.  A  dressmaker  "wishes  to  buy  a  piece  of  silk  which  she 
can  cut  into  patterns  of  either  8,  10,  or  12  yards;  how  large 
must  the  piece  be? 

14.  Find  the  smallest  number  that  can  be  divided  by  12, 
14,  or  16,  and  leave  4  remainder. 

15.  Find  the  largest  number  that  is  contained  in  47  and 
77,  with  a  remainder  of  2. 

16.  What  are  the  least  equal  sums  a  man  can  spend  in 
buying  sheep  at  $3,  calves  at  $12,  cow^s  at  $30,  and  horses 
at  $75  ?     How  much  for  all  ? 

17.  A  teacher  distributes  56  cards  to  one  class,  63  to 
another,  and  77  to  a  third,  giving  the  same  number  to  each 
pupil;  how  many  pupils,  and  how  many  cards  to  each? 

18.  Find  the  least  number  of  cards  that  can  be  equally 
distributed  among  7,  11,  14,  or  22  girls. 

19.  A  kind  lady  has  18  pears  and  33  apples  which  she 
wishes  to  give  to  some  poor  children  in  equal  numbers;  how 
many  can  she  give  to  each,  and  to  how  many  can  she  give? 


CALIFORNIA   SERIES. 


FRACTIONS. 

How  would,  you  divide  5  apples  equally  among  2  boys  ? 
If  you  divide  7  apples  among  2  boys,  how  many  would 
each  receive  ?     Among  3  boys  ?     8  apples  among  3  boys  ? 
5~2=?       5--3=?       7—2=?       7--3=?       8--3=? 
What  does  the  form  of  expression  ^,  |,  f ,  indicate? 

A  Fraction  is  an  indicated  division.  Thus,  the  indicated 
division  of  the  remainder  in  Division  is  a  fraction. 

If  you  divide  1  apple  equally  among  3  boys,  what  part 
does  each  receive  ?  If  you  divide  2  apples  equally  among 
3  boys,  what  part  of  each  apple  does  1  boy  receive  ?  From 
both  apples  how  many  pieces  does  he  receive  ? 

Which  would  he  prefer,  one  piece  from  each  of  the  apples 
or  two  pieces  from  one  apple? 

Then  -g  of  2  apples  is  the  same  as  f  of  1  apple. 

A  fraction,  therefore,  may  be  regarded  as  an  equal  part  or 
as  equal  parts  of  a  unit  or  one.     When  so  regarded: 

(1)  The  denominator  (namer)  names  the  parts  and  shows 
their  size; 

(2)  The  numerator  (numherer)  shows  the  number  of  parts. 

^,         3       three    (number). 
Thus,  -=  .  ,^,    '  , 

8     eighths  (name J. 

In  the  written  expression  is  the  denominator  above  or 
below  the  line  ?  The  numerator  ?  To  what  term  in  division 
does  each  correspond?     (See  pages  43  and  44.) 

The  numerator  and  denominator  are  the  terms  of  the 
fraction. 

The  value  of  a  fraction  is  the  quotient  obtained  by  per- 
forming the  indicated  division. 

A  whole  number,  also  called  an  integer,  may  be  expressed 
in  fractional  form  by  using  1  for  a  denominator.    Thus, 
For  ■},  ^,  read  7  fs,  27  fs. 


ARITHMETIC.  73 

A  combination  of  a  whole  number  and  a  fraction  forms  a 
mixed  number ;  as  5|,  117^. 

A  quotient  in  division,  with  the  remainder  at  the  right 
over  the  divisor,  is  a  mixed  number. 

When  the  dividend  is  equal  to,  or  larger  than,  the  divisor, 
the  indicated  division  is  an  improper  fraction;  for  the  di- 
vision may  be  performed  and  the  result  expressed  as  a 
whole  or  a  mixed  number. 

J-ims,    2  7    3''     4   5    13- 

What  is  the  quotient  in  each  of  these  improper  fractions? 
How  do  you  prove  the  division?     (See  test,  p.  50.) 

EXERCISE  95.    (Written.) 

In  the  following  examples  write  on  your  slate,  in  separate 
columns,  tho  fractions  (proper),  the  improper  fractions,  and 
the  mixed  numbers;  change  the  improper  fractions  to 
mixed  or  whole  numbers,  and  prove. 

1      1    A2l     IJi.  ft     A4.    95J_    iJ_3  IK      9Q1    4JL    2J^ 

■•■•     8i   ^55     4   •  °*     335    ^'-TT?       3     •  ■'■*'•     ^'^85   ^14'    42' 

9      119.        7         .'^U.  Q        41       117      /jQ  3  IR       11      17      29 

^'        7     1    119)   '^17-  ^*     1G45     10   5   ^^25-  •*•"•     14?    517    oS' 

Q      79J     185.    1_7.  TH      1Q11    U.     48.  17       2  GO      50  1       7  . 

o.     <_4,    -LOy,     g   .  J.U.     -1^^505    775    75-  •*• '  •       ii   •,     29   7    ll' 

4.       274  5     .a  11        2  5-|ri3-195  1ft       11^28      917 

fi      4      2_7_     127  19       9Q  5       187      209  1Q      13.0.     5_S      7_1_ 

*'•     83     8   J       5     •  ■'■'^*     '^'-'12?       7     7     11-  ^^'        7     7    1G7    '56- 

fi  12.A    1  ^13.  180  1  q  4  0.1     375     209  On  13_3.  18.     1  Q  9 

u.  J207    ^"157  225-  ^^'  1307    3907    220*  ■^"'       19   5  3  2  7  ^ -•■  "^T2"  * 

7  1442      1  2  5  il_2.  14  907.    lii     175  91  411  473      171 

'•  -"^^^37      .6     7  144-  •'•*•  -^97       3     T   ^  ^  S-  ^^'  ^-^27  ^'47  T^"0^- 

Select  the  first  10  proper  fractions  to  analyze  orally  in  the 
class.     Thus, 

I  is  a  fraction  because ;  8  is  the  clenominator 

because ;  7  is  the  numerator  because . 

EXERCISE  96.    (Oral.) 

Form  improper  fractions  on  your  slate  by  placing  the 
dividends  in  the  first  3  columns  of  dividends,  Exercise  GT, 
as  numerators,  and  their  corresponding  divisors  in  the  divi- 
sor column  as  denominators,  and  bring  into  the  class  for 
oral  work  in  changing  to  mixed  numbers. 


74  CALIFORNIA   SERIES. 

EXERCISE  97.    (Oral.) 
Change  to  improper  fractions: 

1.  4i,  7i  5.  3|,  If.  9.  8|,  7^V  13.  9,^,  8|. 

2.  3f ,  5t.  6.  5t,  llf.  10.  12f ,  7t.  14.  7i^,  7f . 

3.  9i  9i  7.  9f ,  7|.  .  11.  4i  5f .  15.  12f ,  lO^o- 

4.  7|,  8f  8.  5^,  9^^.  12.  6f ,  7i  16.  15|,  5|. 

EXERCISE  98.    (Written.) 
Change  to  improper  fractions: 

1.  4511  72^0-  5.  13A,  17|.  9.1^h\,  l^A- 

2.  109i,  25j\.  6.  85y«5,  63^0-  10.  1041,  i06f. 

3.  58^2^,  193V  7.  49A,  20i|.  11.  781,  49^. 

4.  1401,  14^V  8.  240i,  10.^-  12.  lO^o,  19/o. 

Also  all  mixed  numbers  in  Exercise  95,  and  prove. 

EXERCISE  99. 

JVrite  ten  mixed  numbers  of  your  own  and  reduce  them 
to  improper  fractions.  Also  ten  improper  fractions  and 
reduce  them  to  mixed  numbers. 

To  reduce  to  lower  or  higher  terms. 

If  you  cut  each  half  of  an  apple  into  two  equal  parts, 
what  kind  of  pieces  do  you  get  ?  How  many  4ths  in  each 
half?  Cut  each  fourth  into  2  equal  parts,  and  you  get  what 
kind  of  pieces?  How  many  in  each  fourth?  How  many 
in  f  ?  Cut  each  half  into  3  equal  pieces,  and  you  get  how 
many  pieces  in  all  ?  Name  of  each  piece  ?  How  many  in  i  ? 
Cut  each  third  into  2  equal  parts,  and  you  get  how  many 
pieces  in  all?  How  many  in  each  third?  How  many  in 
I?  Write  these  results  in  a  row  on  your  slates;  thus, 
1 2 4      1 3.      1 ,2      2 4 

"2 4 8?        2 65        3 65        3 6- 

Also,  in  a  second  row  under  these,  write  each  expression 
backward;  thus, 

4 2 1      3 1      2 1      1 — 2 

■g 4 ^5        6 ^5        6 35        6 3* 


ARITHMETIC.  75 

In  the  upper  row,  what  do  you  do,  in  each  case,  with  the 
numerators  and  denominators  of  tlie  fractions  on  the  left  of 
the  sign  (=)  to  get  those  on  the  right?  What  in  the  lower 
row  ?    Which  is  the  larger,  \  or  f  ?    \  or  |  ?    I  or  |  ?    |  or  |  ? 

Multiplying  hoth  numerator  and  denominator  of  a  frac- 
tion hy  the  same  number  is  reducing  the  fraction  to  higher 
terms,  as  in  the  upper  row  abov^e. 

Dividing  both  numerator  and  denominator  of  a  fraction 
by  the  same  number,  is  reducing  the  fraction  to  lower 
terms. 

If  both  numerator  and  denominator  are  divided  by  all 
their  common  factors,  or  their  g.  c.  f.,  the  fraction  is  re- 
duced to  its  lowest  terms. 

If  there  i?  no  common  factor,  the  fraction  is  already  in 
its  lowest  terms. 

Either  of  the  above  processes  does  or  does  not  change  the 
value  of  the  fraction,  and- why?     (See  Prin.  3,  p.  56.) 

EXERCISE  100.    (Oral.) 
Reduce  to  lowest  terms: 

1_§__7__6_  '^6._i_l_2  Ql21fi28  iq3rt36     33 

^'     12314J10-        ^'    9548548-  ^'    36J323   ST*         ^'^'    ¥2"?  To'  To' 

9_9_lJLi8.  fii8.i5.2_0  10      161618  1J.33     8  6o 

^-     15'  21?  24-         "•     30'   30'   30-        ^^'    24'TO'TO-         ■••*•    4T'   oT' To"- 

3i8.2_0.i8.  7S_02j42_4         11_8__20._5_  IK      16304.') 

^'     27'   325   36-  '•     3G'   305   36"        ■'••'■•     185  60'  6  0'         ■'•''•    48'  GO'   54" 

4.-5__8__6_  Qio.42.2.i         19      1_0       7       14  1C      15     1210 

^'     15?  485  48-         °*     35'  49'   28"        ^^'     35'   355   35'         ■■■"•    To'  6""0'  TO* 

EXERCISE   101.    (Written.) 

Form  examples  by  writing  the  smaller  number  in  each 
numbered  couplet  of  Exercise  84  for  a  numerator,  and  the 
larger  for  a  denominator,  and  reduce  to  lowest  terms. 

In  Exercise  86,  select  the  smallest  and  largest  numbers 
of  each  example  for  numerator  and  denominator,  respect- 
ively, and  reduce  to  lowest  terms. 

Also,  reduce  to  lowest  terms  the  proper  fractions  in  Ex- 
ercise 95. 


76  CALIFORNIA   SERIES. 

EXERCISE   102. 

Write  and  perform  1 0  examples  of  your  own  like  those  of 
the  preceding  exercise,  and  bring  to  the  class  for  dictation. 

To  reduce  to  a  common  denominator. 

By  what  number  must  you  multiply  both  numerator  and 
denominator  of  ^  to  change  it  to  6ths  ?  To  8ths  ?  To  lOths? 
Tol2ths? 

f  to  change  it  to  6ths?     To  9ths?     To  12ths?    To  I5ths? 

I  to  change  it  to  8ths?    Tol2ths?    TolGths?    To20ths? 

ftochangeittolOths?    To  15ths?    To20ths? 

Give  results  in  each  case. 

Of  the  above  fractions,  ^,  |,  I,  |,  which  have  you  changed 
to  6ths?     Write  on  your  slate  thus: 

I I     Of  the  denominators  2  and  3,  what  is  6? 

3 6" 

Which  have  you  changed  to  8ths  ?  To  lOths  ?  To  1 2ths  ? 
Of  the  denominators  2,  3,  and  4,  what  is  12? 

Changing  fractions  of  different  denominators  to  equiva- 
lent fractions  having  the  same  denominator  is  reducing  to 
a  common  denominator. 

When  the  new  denominator  is  the  smallest  number  (1.  c. 
m.)  that  can  be  used,  it  is  the  least  common  denominator. 

EXERCISE    103. 

Write  the  fractions  of  each  example.  Exercise  100,  in 
lowest  terms,  numbering  the  examples  as  in  the  exercise. 

Do  the  same  with  the  fractional  parts  of  the  examples  in 
Exercises  97  and  98.  Bring  into  the  class  for  oral  work  in 
reducing  to  least  common  denominator. 

Name,  at  sight,  several  common  denominators  for  each 
example,  and  decide  which  is  the  least  common  denomi- 
nator. 


ETTI^} 


ARITHMET]^.^  77 

EXERCISE  104.    (Written.) 

Change  the  fractions  of  each  example,  Exercise  95,  to 
equivalent  fractions  having  their  least  common  denomi- 
nator. 

EXERCISE   105. 

Write  10  examples  of  your  own,  each  containing  3  frac- 
tions in  their  lowest  terms.    Bring  to  the  class  for  dictation. 


ADDITION   AXD    SUBTRACTION. 

What  kinds  of  objects  can  be  put  together,  or  added? 
What  subtracted? 

Fractions  of  the  same  name  (denominator)  may  be  added 
and  subtracted.     Thus, 

5  apples  and  3  apples  are  hoiv  many  apples? 

5  sevenths  and.  3  sevenths  are  how  many  sevenths f 

Express  the  latter  in  the  written  form ;  tlius,  |-f  |-=^:rr:li. 

EXERCISE    106.     (ORAL.) 

The  fractions  in  the  answers  must  be  proper,  and  im low- 
est terms : 

1        5_1.J ^ 2. ?  Q        3      17  9140 ? 

■•••     8^8        8        8 •  O*     4T^4T        4Ti^4T • 

9      _6 L  HI 8 I 1_ 9  Q      2  1^_i0 5  3  9 

^'   1  sn^  1  sn^isn^is — •  ^'   23      23      23      23 — • 

Q      _7 ]L4_IJL 7_ 9  in  7        17  4        110 

'^'     30         30^^30         30 •  ••■"• 


160    I    160        16  on    16  0 

4.        19      16  24      14     9  11       4  3 4  0._1      5      13   9 

*•     125^^125        125^^125 •  "'■■'••     88        88    I    88    I    88 ' 

5      2_1._I   1_9_L3JL 2_ 9^  lO       14    1 5 I      7      I      f)    9 

fi      A_l_5. 4 3. 9  1Q        8      I    H. 3__L10 9 

"•     9    19        9        9 •  ■'•'-*•     19'ri9        19    I    19 • 

7      1_6 5__l 4  §_— 9  14.       2.0.    I    20 2  2  .9 

'•17        17ri7        17 •  •'■^•21I21        21        21 • 

1R      JL7_|_i7._4_J_i 6_ 9 

^'^'   isn   i8n^i8      18 — • 

We  have  seen  that  fractions  to  be  added  or  subtracted 
must  have  the  same  name,  or  denominator.    What  did  you 


/. 


78  CALIFORNIA   SERIES. 

find  could  be  done  to  fractions  of  different  names  on  page 
76?  -Then  to  add  or  subtract  such  fractions  what  must 
first  be  done  ? 

EXERCISE   107. 

In  Exercise  100,  write  the  fractions  of  each  example  in 
lowest  terms  and  add.  Perform  mentally  as  far  as  possible, 
combining  first  those  most  easily  reduced.     Thus,  in 

Example  1.  |+i=i-=li.     li+f=l|f- 

Also  find  the  difference  between  the  third  fraction  and 
the  sum  of  the  first  and  second  in  each  example. 

If  any  or  all  of  the  numbers  are  mixed  numbers,  add  or 
subtract  the  fractional  parts  first.  In  adding,  reduce  sums, 
if  improper  fractions,  to  whole  or  mixed  numbers  and  add 
the  whole  part  to  the  sum  of  the  whole  numbers. 

In  subtraction,  if  the  fraction  in  the  subtrahend  is  larger 
than  the  fraction  in  the  minuend,  take  one  from  the  whole 
number  of  the  minuend  and  add  it  to  the  fraction  of  the 
minuend,  making  an  improper  fraction;  then  subtract. 

EXERCISE    108.    ^Oral.) 

Give  the  sum  and  the  difference  of  the  mixed  numbers  in 
each  example  of  Exercises  97  and  98. 

EXERCISE    109.    (Written.) 
Add  the  expressions  in  each  example,  Exercise  95. 

EXERCISE   110.    (Oral.) 

Subtract  -f^  from  each  of  the  following;  also  3f : 

10        23        41         36        84        93  102       170 

Subtract  4f  from  each  of  the  following;  also  10|: 

20^        32^        44|        55^        61^  78^        93| 

EXERCISE  111.    (Written.) 
Subtract  \^  from  each  mixed  number  of  Exercise  98. 


ARITHMETIC.  79 

PRACTICAL  WORK  IN  ADDITION  AND  SUBTRACTION. 

EXERCISE  112.    (Oral.) 

1.  Lucy  gives  away  \  of  her  apples  to  Sarah  and  \  to 
Jane;  what  part  has  she  left? 

2.  I  cut  3  pieces  of  cloth  from  3  yards;  the  first  contained 
I  of  a  yard,  the  second  ^,  and  the  third  |;  what  part  of  the 
3  yards  was  still  left? 

3.  John  gave  away  i  and  \  of  his  marbles,  respectively, 
to  two  playmates;  what  did  he  give  to  both? 

4.  From  |  of  a  yard  of  ribbon  I  cut  f\  of  a  yard;  what 
length  of  piece  remained  ? 

5.  Mr.  A.  buys  2f^j  acres  of  land,  and  fences  \\  acres; 
what  remains  unfenced? 

6.  Frank  received  on  Christmas  $24  from  his  father,  $1| 
from  his  mother,  and  $^  from  his  sister;  what  did  he  re- 
ceive in  all? 

7.  How  much  more  from  his  father  than  from  his  mother? 

8.  Than  from  his  sister? 

9.  Than  from  both  mother  and  sister? 

10.  He  spent  %^  for  marbles  and  '^^  for  a  top;  what 
had  he  left? 

11.  George  lives  2|  miles  from  the  school-house;  how 
many  miles  does  he  have  to  go  a  day,  going  and  coming 
once  ? 

12.  William  jumped  8f  feet,  and  John  7|  feet;  how  much 
farther  did  William  jump  than  John? 

13.  Charles  earns  $1^  Monday,  '^1-^  Tuesday,  $lf 
Wednesday,  $1^3^  Thursday,  U\  Friday,  and  $1|  Saturday; 
he  spends  $3|  during  the  week:  how  much  does  he  save? 

14.  I  have  3  pieces  of  carpeting  containing  9f ,  7:j,  and  8|- 
yards,  respectively;  what  is  the  length  of  the  three  pieces 
sewed  together,  allowing  \  yard  for  laps  ? 

15.  I  have  a  journey  of  13^^  miles  to  go;  after  walking 
3|  miles,  how  much  farther  have  I  to  go? 


80  CALIFORNIA   SERIES. 

16.  A  farmer  has  4^  acres  in  orange  trees  and  2%  acres 
in  lemon  trees;  how  much  has  he  in  both? 

17.  How  much  more  in  one  than  in  the  other? 

18.  I  read  3^^  hours  on  Monday,  2^^  hours  on  Tuesday, 
and  3|-  hours  on  Wednesday;  how  many  hours  did  I  read 
in  three  days? 

19.  How  much  longer  on  Monda}^  than  on  Tuesday  or 
Wednesday  ? 

EXERCISE    113. 

Form  5  examples,  like  the  preceding,  in  addition,  and  5 
in  subtraction,  using  the  mixed  numbers  in  the  first  10 
examples,  Exercise  97,  and  bring  to  the  class  for  dictation. 

EXERCISE    114.    (Written.) 

Rewrite  the  first  10  examples  in  general  form  (A  or  B, 
page  28),  before  performing. 

1.  I  paid  $41^  for  a  watch,  $3J  for  a  chain,  and  $|  for  a 
ring;  what  was  my  bill? 

2.  I  raised  on  my  land  last  year  155|  centals  of  wheat, 
76|  centals  of  oats,  and  111|-  centals  of  barley;  how  many 
centals  of  grain  did  I  raise  ? 

3.  From  a  piece  of  land  containing  723|-J  acres,  I  sold 
149-J-l  acres;  what  had  I  left? 

4.  Sold  a  horse  for  $125,  which  was  $13|  more  than  he 
cost  me ;  what  did  he  cost  me  ? 

5.  I  start  on  a  journey,  going  -f^  of  the  distance  the  first 
day,  and  f  the  second  day;  what  part  of  the  journey  have 
I  yet  to  travel  ? 

6.  In  a  certain  school  -^f^  of  the  pupils  are  boys;  what 
part  are  girls? 

7.  A  three-sided  field  has  its  sides  31y\,  46|,  and  59||- 
rods  long,  respectively;  how  far  is  it  around  the  field  ?  Draw 
a  diagram  to  illustrate. 

8.  A  merchant  buys  a  barrel  of  sugar  containing  237^ 
pounds;  he  sells  17-|  pounds  at  one  time,  23|  at  another, 


ARITiniETIC.  81 

and  41|  at  a  third;  how  many  pounds  of  sugar  are  still  in 
the  barrel? 

9.  A  man  traveled  8yV  hours  on  Monday,  9^  J-  on  Tuesday, 
llf^  on  Wednesday,  8^  on  Thursday,  13yV  on  Friday,  and 
lof  on  Saturday;  how  many  hours  did  he  travel  in  all? 

10.  He  traveled  27|  miles  on  Monday,  34f^  on  Tuesday, 
Slg^V  ^iles  on  Wednesday,  and  1791  miles  in  all;  how  far 
did  he  travel  the  last  three  days? 

11.  From  a  piece  of  cloth  containing  108^  yards,  3  pieces 
of  17f,  18^,  and  14|  yards,  respectively,  were  cut;  what 
remained  ? 

12.  A  church  steeple  reaches  lOlyV  ^et  from  the  ground; 
the  roof  of  the  church  is  53^  feet  high;  how  high  does  the 
steeple  rise  above  the  roof? 

13.  Two  men  travel  around  a  pond;  the  first  goes  \  of 
the  distance,  and  the  second  -f^  of  the  distance,  in  one  hour; 
how  much  of  the  distance  has  one  gained  upon  the  other? 

14.  4  piles  of  wood  contain  37y^6-,  41|,  2^^,  and  54^4 
cords,  respectively;  how  many  cords  are  in  the  4  piles? 

15.  One  of  two  stations  is  171^^  miles  east  of  a  certain 
point,  and  the  other  is  235^^^  miles  west  of  the  same  point; 
how  far  apart  are  they? 

16.  How  far  apart  if  both  stations  are  east  of  the  point? 

17.  I  bought  of  one  man  1194  acres  of  land,  of  another 
91^V  acres,  and  of  a  third  75f|-  acres;  what  amount  did  I 
buy  in  all  ? 

18.  A  man  had  ^  of  his  sheep  in  one  pasture,  \  in  another, 
and  ^  in  a  third,  and  the  remainder  in  a  fourth;  what  part 
are  in  a  fourth? 

19.  A  room  is  17yV  feet  long,  and  14|^  feet  wide;  how  far 
is  it  round  the  room?     Draw  diagram. 

20.  Sold  a  horse  for  $11 73^5-  at  a  loss  of  .$7^;  what  did  I 
pay  for  him? 

21.  A  ship  was  53|-  hours  in  sailing  to  a  certain  port, 

and  41f  in  returning;  how  long  was  she  gone? 
6— A 


82  CALIFORNIA    SERIES. 

To  multiply  a  fraction  by  a  whole  number,  or  the  reverse. 

How  many  apples  are  5x2  apples?  2  apples  multiplied 
by  5?  What  are  5X2  thirds?  2  thirds  times  5?  Write 
this  latter  operation  in  proper  form.  Does  it  make  a  dif- 
ference in  the  product  in  multiplication  as  to  which  term 
stands  first? 

EXERCISE  115.    (Oral.) 

Leave  no  improper  fractions  for  answers. 

1.  9Xf  9.  liXll.  17.  -1X21.  25.  31Xf. 

2.  7X|.  10.  i|Xl2.  18.  ^X25.  26.  25XA- 

3.  1X5.  11.  9XtV  19-  fX29.  27.  |X8. 

4.  IOXt^.  12.  7xlf.  20.  fXlS.  28.  t'oX21. 

5.  13X|.  13.  y2-xl8.  21.  fX20.  29.  fX40. 

6.  15Xtt-  14.  13Xi  22.  19XtV  30.  SOXtt- 

7.  fVXlOO.  15.  17X|.  23.  ITXfV  31.  50xf 

8.  yVX9.  16.  llXf.  24.  IXll.  32.  1X91. 

EXERCISE   116.    (Written.) 

Multiply  each  mixed  number,  of  examples  1  to  4,  Exer- 
cise 98,  by  11.  Multiply  the  fractional  part  first,  and  add 
the  product  to  the  product  of  the  whole  part. 

Multiply  examples  5  to  8,  by  13;  9  to  12,  by  17. 

Mixed  numbers  may  be  changed  to  improper  fractions 
before  multiplying,  when  preferable. 

EXERCISE   117.    (Oral.) 

Multiply  the  mixed  numbers  of  examples  1  to  6,  Exer- 
cise 97,  by  3;  7  to  12,  by  5;  13  to  16,  by  7. 

Multiplying  the  numerator  by  a  whole  number  multiplies 
the  fraction  by  that  number.  Why?  (See  Prin.  1,  p.  56, 
Division.)  What  other  operation  does  that  Prin.  say  will 
multiply  the  fraction  ? 

Which  term  of  the  fraction  is  the  divisor  ? 


ARITHMETIC.  83 

In  what  two  ways,  then,  can  yon  mnltiply  a  fraction  by  a 
whole  nnmber? 

EXERCISE    118.    (Oral.) 

In  the  following,  divide  the  denominator  by  the  whole 
nnmber: 

1.  9X^.  6.  9X|f  11-  IIX7. 

2.  7XH-  7.  yVX5.  12.  20xif. 

3.  15Xt|.  8.  i|X8.  13.  |iX6. 

4.  3^0X25.  9.  llXfi  14.  if^Xll. 

5.  7Xff-  10-  5Xt|-  15.  |fX25. 

EXERCISE  119.    (Written  Analysis.) 

See  analysis,  page  36,  Mnltiplication. 

1.  At  -t?!  a  cord  what  will  25  cords  of  wood  cost? 

2.  Sold  19  yards  of  cloth  at  $2f  a  yard;  find  the  whole 
selling  price. 

3.  What  cost  160  acres  of  land  at  $65^? 

4.  A  barrel  contains  3H  gallons;  how  many  gallons  in  12 
barrels  ? 

5.  What  will  175  centals  of  wheat  cost  at  ^Ijo-  a  cental? 

6.  How  far  can  I  walk  in  11  honrs,  at  the  rate  of  3-y- 
miles  an  honr? 

7.  If  a  certain  number  of  shoes  can  be  sewed  in  ISy^- 
hours  on  12  machines,  how  long  will  it  take,  using  1  ma- 
chine ? 

8.  If  I  can  copy  12^  pages  in  one  day,  how  many  pages 
can  I  copy  in  6  days? 

9.  12  men  buy  a  mill  together,  each  paying  $728|-:  what 
is  the  cost  of  the  mill? 

10.  One  rod  contains  5^  yards;  how  many  yards  are  in 
80  rods? 

EXERCISE    120. 

Write  10  examples  of  your  own,  like  the  preceding  exer- 
cise, using  5  examples  each  from  Exercises  115  and  118. 


84  CALIFORNIA   SERIES. 

EXERCISE  121.    (Oral  Analysis.) 

1.  At  $|-  a  roll  what  will  10  rolls  of  butter  cost? 

2.  What  cost  9  yards  of  cloth  at  .$f  per  yard? 

3.  What  cost  5  rings  at  $2^  apiece? 

4.  How  many  centals  of  grain  will  12  bags  hold,  they 
averaging  1^  centals  to  a  bag? 

5.  What  does  a  man  earn  in  a  week,  at  $1|  a  day? 

6.  If  1^  pounds  of  butter  pay  for  a  yard  of  cloth,  how 
many  pounds  of  butter  will  it  take  to  pay  for  16  yards  of 
cloth? 

7.  What  cost  a  dozen  oranges  at  2^  cents  each? 

8.  I  gave  7  boys  $f  each ;  how  much  to  all  ? 

9.  A  wheel  turns  2|-  times  in  going  a  rod;  how  many 
times  Avill  it  turn  in  going  18  rods? 

10.  Find  the  price  of  13  sacks  of  wheat  at  $lf  each. 

Perform  the  work  of  the  3  upper  rows  in  Exercise  67, 
reading  ■§  of  15,  -J  of  29,  etc. 

I  of  29=?     9f  are  how  many  thirds?     i  of  29=\K 

i  of  44=:  ?     6|-  are  how  many  sevenths  ?     -f  of  44=4^. 

f  are  how  many  times  -g?  If  -g  of  15  is  5,  f  of  15  is 
what  ?     If  i  of  29  is  -2/,  f  of  29=what  ?     f  of  44=  ? 

Write  these  on  your  slate,  in  a  row;  thus, 

I  of  15=10,     |of29=^S    |.of44=if^. 

Below  these  write  in  a  row,  with  results: 

1X15=      ,         1X29=      ,        fX44=      . 

Compare  answers.  "Of,"  between  fractions,  is  therefore 
equivalent  to  what  sign? 

EXERCISE   122.    (Written.) 
Write  and  find  f  of  each  dividend  in  the  upper  row.  Exer- 
cise 67.     In  cases  similar  to  the  first,  write  thus, 

5 

;|of;p=io. 

f  of  numbers  in  the  second  row;  f  in  the  third;  f  in  the 
fourth. 


ARITHMETIC.  85 

EXERCISE  123.    'Oral  Analysis.) 

If  1  dozen  eggs  cost  30  cents,  what  will  -|  of  a  dozen  cost? 

Model. — }4  dozen  will  cost  I3  of  30  cents  =  10  cents.  %  dozen 
will  cost  2x10  =  20  cents. 

1.  What  cost  f  yard  of  cloth  at  20  cents  a  yard  ? 

2.  Bought  I  of  a  cord  of  wood  at  -$8  a  cord;  what  was 
my  bill  ? 

3.  Sold  y^g-  of  an  acre  of  land  at  $48  an  acre;  what  did  it 
cost? 

4.  Sold  y^Q-  of  a  ton  of  hay  at  $8  a  ton :  what  did  I  receive  ? 

5.  From  a  piece  of  cloth  containing  55  yards,  -fi  of  it  was 
cut;  how  many  yards  were  cut? 

6.  Of  63  children  in  a  certain  district,  -|  attend  school; 
how  many  attend? 

7.  Bought  f  of  a  yard  of  ribbon  at  20  cents  a  yard;  what 
did  the  ribbon  cost  me? 

8.  A  row  of  trees  contains  28  trees;  how  many  trees  are 
in  \  the  row? 

9.  In  a  certain  school  containing  56  scholars,  |  are  in  the 
first  grade  and  |  in  the  second ;  how  many  in  each  ? 

10.  Received  $2  for  a  day's  work  of  10  hours;  what  did 
I  receive  per  hour? 

11.  If  f  of  10  questions  are  missed,  how  many  are  missed  ? 

To  multiply  a  fraction  by  a  fraction. 

The  upper  lines  divide  the 
whole  line  into  how  many  ItV 
parts?  The  lower  lines?  Into  how  many  parts  do  the 
lower  lines  divide  each  fifth?  Then  -g  of  \=^Yo'^  or  (of=X) 
J-  X  J- 1- 

,rof|=?     fofi==?     |of|=?     iof4=.?     |of4=? 

Use  the  sign  ( X ) ,  wTiting  both  ways.  How  can  you  ob- 
tain your  new  numerator  in  each  case  from  those  of  the 
multiplicand  and  multiplier?    The  new  denominator? 


1 

1 

3. 

4 

5 

1 

5 

f 

-?• 

t; 

1  1 

1 

1     1     1 

1        1        1 

1        1 

1     1 

1T5 

1 

1    i    i 

1        1        1 

1        1 

1     1 

86  CALIFORNIA   SERIES. 

EXERCISE  124.    (OralJ 


1. 

3  v2 

9. 

3     nf    -^ 
20   Ol    10- 

17. 

3v    3 

25. 

2.  of -2 
9    '-'1    5- 

2. 

5  V  5 

10. 

7    nf  4 
3  0   01   5- 

18. 

2.  of  2. 
5   ^^   9- 

26. 

2.  of  2. 

3. 

^oftV 

11. 

^  nf  6 

4   01  y. 

19. 

5v    5 
¥/\12- 

27. 

2  0/^5- 

4. 

i\  of  I 

12. 

10  of  1 0 

11  ^1    11- 

20. 

4\/    5 
7  /\14- 

28. 

■3-V   '"^ 

5. 

i  X  T  6"- 

13. 

7  Ag- 

21. 

9/\3- 

29. 

5  nf  '"^ 
T  01  T4 

6. 

J?_  of  « 

11    ^^    7- 

14. 

8    V    8 
11/^11- 

22. 

8  of     8 

9  Ol    n- 

30. 

A  of  J 

7. 

5     nf  1  0 
T3  01  TT- 

15. 

15V/1 
1  6  /\  2  • 

23. 

4    V-^ 
13/\3- 

31. 

lx|. 

8. 

2  of     8 

3  01    11- 

16. 

loff 

24. 

_8_  of  A 
11    ^1    9- 

32. 

Axi 

What  is  ^  of  9  apples  ?    |  of  9  apples  ? 
What  is  ^  of  9  tenths?    f  of  9  tenths?    Write  the  last 
two  expressions  in  full. 

3 

^  of  /o=tV     (See  Exercise  122.) 
I  of  y%  are  ^Xt^-^I-     (^ee  Exercise  115.) 
5  3 

Hence,  in  writing  the  work,  shorten;  thus, ^  of  ;^=^f . 

5 
This  method  of  work  is  called  cancellation. 

A  common  factor  of  any  numerator  and  denominator,  in 
multiplying,  may  be  canceled  by  Prin.  3,  p.  56.     Thus, 

J      '^ jo_  dividing  4  and  14  by  2,  and  9  and  15  by  3, 


^/\i.*        2  1 J 

3     7 


before  multiplying. 


What  is  Prin.  3,  referred  to? 


■■■•  5  6  Ayr- 

-^^  2"5/\"39"- 

q  14   of   25 

'^'  1  5   Ol    2T- 

4  11  V.7 

K        5  7    V  1  2 

'^^  108/^19- 

"•  105/^120- 


EXERCISE   125.    (Written.) 

7         8  5      p,f   8  1 
'  •    TO  8    01  -9  5"- 

13. 

3  2  \/  2  5 
75/^48- 

O        8  4     of  1  7 
O-     119   01   2  4- 

14. 

3  2  5  \/    1 
7      /\25 

Q      5  5  v  1  4  1 
^'    81/^143- 

15. 

3     \/32 
160 /\    9 

10      •'?  9    of  4  <^ 
±U.     6  0   ^1    6  5- 

16. 

7   V  24 

11       2  1  V  1  S 

17. 

1  5  0  V  3 
151  /\5. 

1  9       1  9   nf  1  '"^ 
1^.     2  0   ^1    2  0- 

18. 

1  9  V    7 

ARITHMETIC.  87 

EXERCISE  126.   (Written.) 

Multiply  together  the  mixed  numbers  of  each  example 
in  Exercises  97  and  98. 

EXERCISE   127.    (Oral.) 

Write  on  your  slates,  in  their  lowest  terms,  with  "of"  or 
sign  (X)  between,  the  fractions  of  each  example  in  Exer- 
cise 100.     Bring  to  the  class  to  multiply. 

EXERCISE   128.    (Written  Analysis.) 

1.  What  cost  18|  yards  of  carpet  at  $22^  a  yard? 

2.  How  far  will  a  railroad  train  go  in  2 If  hours  at  the 
rate  of  19f  miles  an  hour? 

3.  What  is  the  value  of  a  pile  of  wood  containing  45yV 
cords,  at  $13f  a  cord? 

4.  At  %l  a  pound  what  wall  9f  pounds  of  coffee  cost? 

5.  Find  the  price  of  21^  yards  of  ribbon  at  12^  cents. 

6.  What  is  the  weight  of  llf  barrels  of  flour  averaging 
197|  pounds  each? 

7.  At  $4^  a  barrel  what  will  84  barrels  of  flour  cost? 

8.  What  is  the  cost  of  building  a  fence  29|  rods  long  at 
$2^  a  rod  ? 

9.  A  certain  river  flows  7yV  niiles  an  hour;  how  far^  will 
a  boat  float  on  it  in  20|  hours  ? 

10.  A  wind  blowing  27|-  miles  an  hour  blows  how  far  in 

9^  hours? 

EXERCISE    129. 

^^"rite  10  examples  similar  to  the  preceding,  using  the  last 
10  examples  of  Exercise  97. 

To  divide  a  fraction  or  a  mixed  number  by  a  whole  number. 

A\^rite  the  following  with  answers  on  your  slate: 
9  apples-^3=  18  dollars-^6= 

9  tenths--3=:  18  twenty-fifths --6= 

12  thirteenths--3=  (2^)  30*^ elevenths ^6^ 


88  CALIFORNIA   SERIES. 

25  cents -i-5= 

(3^)  25  eighths— 5-3 

(2f )  20  sevenths— 5= 

Dividing  anything  by  2  is  taking  what  part  of  it?  By  4? 
By  7?  Taking  ^  of  anything  is  the  same  as  multiplying  by 
what?  Then  dividing  by  2  is  the  same  as  multiplying  by 
what?     By  4? 

Write  on  your  slate  the  following: 


3 

5 

fi 

8 

1  1 

14 

16 

9 

T 

9 

1  3 

1  'J 

5 

15 

17 

2 

Divide  each  by  2  by  taking  -|  of  it;  by  4  by  taking  ^  of 
it;  by  7  by  taking  ^  of  it. 

EXERCISE   130.    (Oral.) 

Divide  the  following  by  each  number  in  turn  from  2  to  9, 
choosing  the  better  of  the  two  methods  above: 

7  __9_  91.  A  1  4  _§_  Kl_ 

11  10  ^7  7  2  9  11  "^7 

When  the  dividend  is  a  mixed  number  whose  whole  num- 
ber is  larger  than  the  divisor,  divide  as  in  whole  numbers, 
reduce  the  remainder  to  an  improper  fraction  and  divide  it. 
Thus,  4  '^'^  ^-T='^i  ^"'^^  -^T5  ^^'  fj  ove7\     4  ^'^  f=T-     ^'i^s.  4t' 

EXERCISE  131.    (Written.) 

Select  10  expressions,  either  improper  fractions,  proper 
fractions,  or  mixed  numbers,  reducing  mixed  numbers  to 
improper  fractions,  and  divide  by  numbers  that  are  factors 
of  the  numerators;  also  10  others,  with  divisors  that  are  not 
factors  of  the  numerators. 

EXERCISE  132.    (Oral  Analysis.) 
Models  on  p.  46,  Division. 

1.  If  7  dozen  eggs  cost  ^If,  what  are  eggs  a  dozen? 

2.  If  5  pounds  of  butter  pay  for  11^  yards  of  cloth,  how 
many  yards  does  1  pound  pay  for  ? 


ARITHMETIC.  89 

3.  When  I  pay  $42 1  for  5  cords  of  wood,  what  is  the  price 
per  cord  ? 

4.  At  $5  a  yard  how  many  yards  of  silk  can  I  get  for  $18|  ? 

5.  Allowing  9  hours  a  day,  how  many  days'  work  will  47^ 
hours  make? 

6.  If  1  man  can  do  a  piece  of  work  in  9-^  days,  in  what 
time  can  7  men  do  it  ?     9  men  ? 

7.  When  5  gold  rings  cost  $11^,  what  are  they  apiece? 

8.  At  $9  a  cord  how  many  cords  of  wood  can  he  bought 
for  $20i? 

9.  At  $2  a  day  how  long  will  it  take  a  man  to  earn  '$12|? 

10.  For  %22\  I  bought  cloth  at  %o  a  yard;  how  many 
yards  did  I  buy? 

To  divide  a  whole  number  or  a  fraction  by  a  fraction. 

How  man}"  times  are  3  dollars  contained  in  15  dollars? 
In  17  dollars?     In  2  dollars? 

How  many  times  are  3  fifths  contained  in  15  fifths?  In 
17  fifths?     In  2  fifths? 

But  ^=3  and  ^-=^.     Hence  3,  or  -U,--|=5 ;  3|,  or  -U, 

•  3 JJ7 P;2.         2    .    3 2 

•  5—   3  ^3-       5'-    5 3- 

f  are  how  many  15ths?     f  are  how  many  15ths?  .  Then 

2..    3 \Q__. 9_ 10     r»r   1  1 

3    •    5 15    •    15 ~9~;  ^^    ^'^• 

EXERCISE   133.    (Written.) 
Divide  each  of  the  following  by  |.  f ,  and  f  in  turn  : 

11  91  11  11  2.  5  1  7^ 

^  8  ^4  T2'  -"-g"  3  TT  "6  '  "^ 


INVERTING  THE  DIVISOR. 

By  Prin.  1  and  2,  p.  56,  Division,  and  their  explanations, 
we  find  that  the  smaller  tliB  divisor  the  larger  the  quotient. 
Di^'iding  by  f ,  then,  will  give  a  quotient  5  times  larger  than 


90  CALIFORNIA   SERIES. 

dividing  by  3,  because  f  is  5  times  smaller  than  (or  ^  of )  3. 
But  dividing  any  immber  by  3  is  taking  ^  of  it;  therefore 
dividing  by  f  is  5)<-i,  or  -f  of  it;  or  multiplying  by  f  (f  in- 
verted). Thus,  i^-:-f==itXf.  With  fractions  of  different 
denominators  this  is  the  shorter  process,  except  in  cases 
where  the  numerator  and  denominator  of  the  dividend  are 
respectively  divisible  by  the  numerator  and  denominator  of 

the  divisor;  as  6 

JtJgr  .  ^ 6 1 1 

5 

EXERCISE    134. 

Repeat  Exercise  133  orally  by  this  method. 
Also  divide  the  first  mixed  number  by  the  second  in  each 
example,  Exercise  97. 

EXERCISE   135.    (Written  Analysis.) 

1.  At  $2f  each  how  many  chairs  will  $28y%  buy? 

2.  At  $3^  a  day  how  many  days  must  a  man  work  to 
earn$245f? 

3.  How  long  will  it  take  a  tree  to  grow  30  feet  high  at  an 
average  of  3y\-  feet  a  year? 

4.  Allowing  9|  yards  to  a  dress,  how  many  dresses  will 
126f  yards  of  cloth  make? 

5.  2SJ-  dozen  buttons  will  be  sufficient  for  how  many 
dresses,  allowing  2i  dozen  to  a  dress? 

6.  A  man  divided  458^  A.  of  land  among  his  sons,  giving 
each  IHyV  A.;  how  many  sons  had  he? 

7.  I  divide  a  pole  3  yards  long  into  divisions  -I-  of  a  yard 
long;  how  many  divisions? 

8.  If  you  take  If  feet  to  a  step,  how  many  steps  will  you 
take  in  going  16|  feet? 

9.  A  man  digging  a  ditch  38|  feet  long,  digs  7^  feet  a 
day;  how  many  days  will  it  take  him? 

10.  A  cubic  foot  of  air  weighs  1|  ounces;  how  many  cubic 
feet  of  air  will  weigh  16  ounces  ? 


ARITHMETIC.  91 

EXERCISE   136. 

In  the  following  complex  fractions,  perform  indicated 
operations: 

9"  3  '-'^  4  ^4  Ag- 

in the  following,  multiply  each  expression  of  the  complex 
fraction  by  the  1.  c.  m.  of  the  denominators  of  all  the  frac- 
tions above  and  below  the  main  line,  combining  as  you  go. 
Thus,  in  5,  6  is  the  1.  c.  m.  of  2,  8,  and  6.  6xi==3,  6Xi= 
2,  2+3=5;  6xi=l;  f=5.     Work  mentally. 

71  213  Ql  1-1-2 3. 

Q       *  '3  10       S    1^4  11       "JS  19       2~r3         4 

16  6        4  ^4  12 


191  ^71  ^^1  r^9i 

^'^'  loo  ^*-  loo  ^^'  loo  ^^'  100 

66|  16f  87i  83^ 

100  100  100  100 

To  find  what  part,  or  fraction,  one  number  is  of  another. 

What  is  i  of  7?  -fofV?  f  of  7?  yV  of  15?  t\  of  15? 
A  of  15? 

1  is  what  part  of  7?  2  what  part  of  7?  What  part  of  7 
is  3? 

1  is  what  part  of  15?  What  part  of  15  is  4?  8  what  part 
of  15? 

The  numbers  7  and  15,  of  which  you  are  finding  parts, 
are  found  where  in  the  resulting  fractions? 

AVhere  are  the  numbers  which  are  parts  of  7  and  15  found 
in  the  results? 

Hence  we  see  that,  to  find  Avhat  part  or  fraction  one  num- 
ber is  of  another,  we  form  a  fraction  by  placing  the  number 
that  is  a  part  as  the  numerator  and  the  number  of  which 
it  is  a  part  as  the  denominator.  The  fraction  thus  formed 
should  be  reduced  to  lowest  terms,  when  not  so. 


92 


CALIFORNIA   SERIES. 


EXERCISE   137.    (Oral.) 


AVhat  fraction — 

1. 

of  20  is  8? 

9. 

of  24  is  21? 

17. 

is  8  of  20? 

2. 

of  21  is  9? 

10. 

of  27  is  20? 

18. 

of  27  is  20? 

3. 

is  18  of  32? 

11. 

of  20  is  19? 

19. 

is  8  of  32? 

4. 

of  45  is  15? 

12. 

is  20  of  32? 

20. 

is  12  of  16? 

5. 

is  12  of  18? 

13. 

of  21  is  14? 

21. 

of  29  is  27? 

6. 

of  25  is  20? 

14. 

is  15  of  25? 

22. 

is  33  of  44? 

7. 

is  18  of  24? 

15. 

is  24  of  25? 

23. 

of  30  is  25? 

8. 

is  13  of  20? 

16. 

of  30  is  25? 

24. 

of  24  is  18? 

To  find  the  whole  when  a  part  is  given. 

How  does  \  of  an  apple  compare  in  size  with  f  ? 

1  with  I?    i  with  I?'    iwithf? 

What  is  i  of  9 ?  |  of  9 ?  ^  of  9,  or  3,  is  what  part  of  |  of 
9,  or  6? 

If  6  is  f  of  some  number,  i  of  that  number  is  what  part 
of  6?    If  i  is  3,  fare  what? 

EXERCISE  138.   (Written.) 
18  is  f  of  what  number? 
Model.— If  18  is  §,  i  is  J  of  18,  or  9;  §  are  3x9,  or  27. 


1.  125  is  #  of  what  number? 


6.  642  is  I^J  of  what? 


2. 

144  is  A '' 

u 

a 

? 

7. 

840  is  iV 

u 

u 

? 

3. 

321  is  f  " 

a 

u 

? 

8. 

59  is  i/ 

(; 

u 

? 

4. 

45is-i-  " 

a 

u 

? 

9. 

189  is  y% 

a 

u 

? 

5. 

540  is  ^2 " 

u 

L(. 

? 

10. 

910  is ;; 

u 

ii. 

? 

EXERCISE  139.    (Oral.) 


1.  16  is  ^  of  what? 


2. 

49  is  ,\    " 

ii 

? 

3. 

32is|     " 

ii 

? 

4. 

44isTVo" 

ii 

? 

5. 

28  is  i     " 

ii 

? 

6. 

/    ^      IS       J    y    Q 

ii 

? 

7.  20isi     of  what? 

8.  19is-3-Vo"      "     ? 

9.  64  is  I     " 

10.  50  is  i     " 

11    12  is     3    t< 

11.  ±^  l^    1  oO 

12.  25  is  f     " 


ARITHMETIC. 

93 

13. 

3G  is  f     of  what? 

17. 

33  is  \l    of  what? 

14. 

14isTVo"       "     ? 

18. 

144  is  -V^    "       "     ? 

15. 

25  is  i      "       "     ? 

19. 

108isx%    "       "    .? 

16. 

28isf      "       "     ? 

20. 

/  is  loO 

PRACTICAL  WORK  IN  FRACTIONAL  ANALYSIS. 


General 
Forms. 


G.— What  is  1  of  16  ? 
H. — 12  is  ^  of  what  number  ? 
I. — 12  is  what  part  (fraction)  of  16  ? 


EXERCISE    140.    (Written.) 
Write  the  following  20  examples  in  proper  general  form 
before  performing,  and  analyze: 

1.  A  man  sold  a  watch  for  $36,  which  was  f  of  what  it 
cost  him;  what  did  it  cost? 

2.  A  broker  having  $875  lost  $175  in  speculating;  what 
part  of  his  money  did  he  lose  ? 

3.  A  stock-raiser  sold  250  sheep,  which  were  |  of  all  he 
had;  how  many  had  he? 

4.  I  bought  a  horse  for  $1575  and  sold  it  for  f  of  the  cost; 
what  did  I  get  for  it  ? 

5.  I  of  a  ranch  is  worth  $12300;  what  is  the  whole  worth  ? 

6.  What  is  -^  of  the  above  ranch  worth? 

7.  What  part  of  it  is  worth  $10250? 

8.  At  $7^  a  ton  what  is  f  of  a  ton  of  hay  worth  ? 

9.  What  is  |  of  an  acre  of  land  worth  if  |  of  it  is  worth 
$75? 

10.  A  man  owning  f  of  a  mill  sells  f  of  his  part  for  $5760; 
what  is  the  value  of  the  mill  at  that  rate? 

11.  What  is  the  value  of  the  part  he  has  left? 

12.  Bought  a  watch  for  $65  and  sold  it  at  a  gain  of  $5; 
what  fraction  of  the  cost  did  I  gain  ? 

13.  A  man  having  a  journey  of  248  miles  to  perform  goes 


94  CALIFORNIA   SERIES. 

31  miles  the  first  day;  wliat  part  of  tlie  journey  is  that? 
What  part  has  he  left  to  walk  ? 

14.  A  farmer  sold  108  acres  of  land,  which  was  ^q  of  his 
whole  ranch;  how  large  was  his  ranch? 

15.  After  selling  -f  of  my  sheep  I  have  1200  left;  how 
many  had  I  at  first? 

16.  At  $5|  a  yard  what  will  f  of  a  yard  of  cloth  cost? 

17.  I  buy  a  place  for  $2325  and  pay  $1800  down;  what 
part  of  the  money  do  I  still  owe  ? 

18.  A  man  earns  $1575  a  year  and  spends  f  of  it;  what 
does  he  save? 

19.  A  ship  having  320  tons  of  coal  on  board  sprung  a 
leak,  and  128  tons  were  thrown  overboard;  what  part  of  the 
coal  was  lost? 

20.  -^  of  an  army  was  lost  in  battle  and  8800  men  were 
left;  how  many  men  were  in  the  army? 


ORAL  REVIEW  IN  FRACTIONS. 

EXERCISE    141. 

1.  A  man  gave  ^  of  a  dollar  to  John,  twice  as  much  to 
Eddie,  and  half  as  much  to  Elmer  as  to  the  other  two; 
what  did  he  give  to  all? 

2.  If  he  gave  the  remainder  of  the  dollar  to  Peter,  what 
did  Peter  get? 

3.  Daniel's  kite  string  is  31^  feet  long,  and  he  ties  a  piece 
^  as  long  to  it;  how  long  is  it  now? 

4.  Joseph's  kite-string  is  f  as  long  as  Daniel's  was  at  first; 
find  the  length  of  Joseph's  kite-string. 

5.  Elmer  can  walk  2-|  miles  an  hour,  and  Charlie  3^ 
miles;  how  far  can  both  walk  in  5  hours? 

6.  Katie  and  Nellie  have  10  examples  to  work;  it  takes 
them  8f  minutes  each  to  perform  an  example;  how  long 
will  it  take  both  to  perform  the  10  examples? 


ARITHMETIC.  95 

7.  Mamie,  by  devoting  |  of  an  hour  to  each  lesson,  studied 
2|  hours;  how  many  lessons  had  she  to  get? 

8.  Agnes  devotes  2^  hours  a  day  to  study,  1^  hours  to 
music,  and  1  hour  to  sewing;  what  does  she  spend  on  all  in 
5  days? 

9.  Four  girls,  Bell,  Mell,  Nellie,  and  Susie,  agree  to  lay 
b}^  To  of  ^  dollar  each  week  for  the  Sunday  school ;  what 
do  they  all  lay  by  in  20  weeks? 

10.  Angle  finds  that,  by  working  l-f  hours  on  her  dress 
each  day,  she  can  make  it  in  5  days;  how  many  hours  does 
it  take? 

11.  Edith  and  Hazel  have  each  $1^;  they  agree  to  buy 
in  equal  shares  a  book  costing  %1\  as  a  Christmas  present 
for  their  mother;  what  has  each  left? 

12.  Florence  and  Leona  together  lack  i  of  a  dollar  to 
buy  a  picture  worth  $1^;  if  each  has  the  same  sum,  what 
has  each? 

13.  Hattie  and  Mary  give  ■§  of  an  apple  to  each  of  two 
playmates  and  divide  the  rest  equally  between  themselves; 
what  part  has  each? 

14.  Susie  receives  jq-  of  a  dollar  a  day  from  her  mother 
for  work;  she  Welshes  to  buy  a  dress  worth  .^6^  and  a  hat 
worth  $3^  at  the  close  of  the  school  term;  can  she  do  it, 
counting  the  time  20  weeks  of  5  days  each? 

15.  Antone  drives  the  cows  to  pasture  in  the  morning  and 
John  gets  them  at  night;  if  the  distance  from  their  house  to 
the  pasture  is  |  of  a  mile,  how  far  do  both  travel  in  1  week? 

16.  Marvin  lives  2|  miles  from  the  school -house;  if  it 
takes  him  5^  minutes  to  go  ^  of  a  mile  how  long  is  he  in 
going  to  school  ? 

17.  It  takes  Willie  1^  minutes  to  distribute  the  copy- 
books to  40  pupils;  suppose  each  scholar  were  allowed  to 
get  his  own  copy-book,  taking  \  minute,  how  much  time 
would  be  lost? 

18.  Joseph  finds  he  has  20  pages  of  his  grammar  to  learn 


96  CALIFORNIA   SERIES. 

to  meet  the  requirements  of  his  class;  if  he  learns  1^  pages 
a  day,  how  long  will  it  take  him  ? 

19.  Alfred  and  Charles  together  have  60  cents;  Charles 
has  -J  as  much  as  Alfred ;  what  has  each  ? 

20.  h  of  A's  money  is  -g  of  B's,  and  together  they  have 
$75;  what  has  each? 

21.  At  $4-  a  yard  how  many  yards  of  cloth  can  I  get  for 
$12? 

22.  How  many  sacks  of  potatoes  at  $1^  a  sack  will  pay 
for  a  barrel  of  sugar  at  $18|? 

23.  At  I  of  a  dollar  a  pound,  how  many  pounds  of  coffee 
are  worth  $15|? 

24.  A  can  do  a  piece  of  work  in  3  days  and  B  the  same 
in  6  days;  what  part  can  each  do  in  1  day?  How  long  will 
it  take  both  together  to  do  the  work  ? 

25.  I  deposited  $32y^Q-  in  the  bank,  which  was  -^  of  what 
I  had  there  already;  how  much  had  I  there? 

26.  1  man  builds  a  barn  in  8|-  days;  how  long  will- it 
take  4  men? 

27.  John  can  do  a  piece  of  work  in  8  days,  and  Bertie  in 
12  days;  how  long  will  it  take  both  to  do  it? 

28.  A  pole  is  i  in  the  mud,  |  in  the  water,  and  10  feet 
above  the  water;  how  long  is  it? 

29.  A  boy  has  52  eggs  in  his  basket;  what  are  they  worth 
at  21  cents  a  dozen? 

30.  A  can  dig  a  ditch  in  6  days,  B  in  8  days,  and  C  in  12 
days;  in  what  time  can  all  do  it? 

31.  At  1^  cents  apiece,  how  many  oranges  will  pay  for  11 
yards  of  print  at  9  cents  a  yard  ? 

32.  Bought  90  centals  of  wheat  at  ^Ijo"  ^  cental;  if  I 
give  5  20-dollar  pieces  in  payment,  what  change  do  I  re- 
ceive? 

33.  At  $1|-  a  day,  what  does  a  man  earn  in  5  weeks? 

34.  How  many  oranges  can  I  buy  for  105  cents  at  1| 
cents  apiece? 


ARITHMETIC.  97 

35.  John  had  17  marbles,  which  were  5  less  than  W  of 
James's;  how  many  had  James? 

36.  A  can  do  a  piece  of  work  in  10  days,  C  in  12  days, 
and  B  in  15  days;  in  what  time  can  all  do  it? 

37.  In  what  time  can  A  and  B  do  it?     A  and  C?     B 
and  C? 


WRITTEN  REVIEW  IN  FRACTIONS. 

EXERCISE   142. 

1.  At  $lf  a  yard,  how  many  yards  of  cloth  can  I  get  for 
$9|? 

2.  Bought  of  one  man  11^  acres  of  land  at  -1^37^,  and  of 
another  17f  acres  at  $42;  how  many  acres  had  I  and  how 
much  did  I  pay  for  the  whole  ? 

3.  If  43^  yards  of  silk  cost  $108|,  what  must  I  pay  for 
12|  yards  at  the  same  price? 

4.  Multiply  x%  of  9|  by  4  of  2i 

5.  A  man  having  175^  acres  of  land  sold  -§  of  it  at  one 
time,  and  \  at  another;  what  is  the  remainder  worth  at  $45 
an  acre? 

6.  Sold  20  dozen  eggs  at  %\  a  dozen,  and  received  in  pay- 
ment butter  at  %\  a  pound;  how  many  pounds  did  I  receive? 

7.  Sold  my  farm  for  $2250,  which  was  f  of  its  value; 
what  was  its  value  ? 

8.  How  many  coats  can  be  made  from  175|  yards  of 
cloth,  allowing  2|  yards  to  a  coat? 

9.  The  length  of  a  room  is  17f  feet,  and  the  width  is  12| 
feet;  what  will  be  the  cost  of  a  moulding  around  it  at  3^ 
cents  a  foot? 

10.  How  many  pounds  of  butter  at  22^  cents  a  pound 
will  pay  for  18|  pounds  of  sugar  at  12  cents  a  pound? 

11.  A  sold  f  of  his  farm  of  475  acres  to  B,  and  B  f  of  his 

part  to  C;  how  many  acres  did  B  sell? 
7— A 


98  CALIFORNIA    SERIES. 

12.  A  man  contracts  to  do  a  job  in  60  days;  how  much 
of  the  work  should  be  done  in  22^  days  ? 

13.  A  lady  has  $63|  in  her  purse;  she  spends  $17-|  for 
a  shawl,  $3|-  for  cloth,  $7^  for  a  bonnet,  and  $5^  for  lace; 
how  much  has  she  left? 

14.  There  are  5^  yards  in  a  rod;  how  many  rods  are  in 
104f  yards? 

15.  How  many  yards  in  820  rods? 

16.  Sold  wheat  for  $517y^o-,  gaining  $27f  on  the  cost; 
what  did  I  pay  for  it  ? 

17.  A  man  owning  7^  acres  of  land,  divided  it  into  house 
lots  containing  ^V  acres  each:  how  many  lots  did  he  make? 

18.  If  $f  buys  a  yard  of  cloth,  how  many  yards  will  $7^ 
buy? 

19.  If  3|  pounds  of  coffee  costs  99  cents,  what  will  |  of  a 
pound  cost? 

20.  How  many  tons  of  hay  are  in  19  loads,  each  contain- 
ing yf  of  a  ton? 

21.  What  is  the  price  per  yard,  when  7-|  yards  of  cloth 
cost  $14? 

22.  A  man  owning  |  of  a  mill  sold  fV  of  his  share  to  one 
man  and  y\  to  another;  what  had  he  left? 

23.  A  has  324  head  of  cattle,  and  y^g  of  his  herd  is  -^  of 
B's;  how  many  has  B? 

24.  A  lot  of  goods  was  sold  for  $4774,  of  which  A  owns 
tVj  ^  A?  ^^^^  C  ^^6  remainder;  find  the  money  each  should 
receive. 

25.  A  tailor  wishes  to  put  2^  yards  of  cloth  into  a  coat, 
2-^  into  a  pair  of  pants,  and  |  into  a  vest;  how  many  suits 
can  be  made  from  a  piece  of  cloth  containing  60^  yards, 
and  how  many  vests  from  the  remainder? 

26.  A  can  do  a  piece  of  work  in  J  3  days,  and  B  in  14 
days;  in  what  time  can  both  do  it? 

27.  When  5^  centals  of  wheat  cost  $6^,  how  many  cen- 
tals can  be  bought  for  $12 J^-? 


ARITHMETIC.  99 

28.  A  man  having  sold  f  of  his  hogs,  and  lost  i  by  dis- 
ease, had  150  left;  how  many  had  he  at  first? 

29.  Find  the  whole  value  of  127-i  centals  of  wheat  at  $11- 
a  cental,  18  centals  of  oats  at  $1^  a  cental,  and  75  centals 
of  barley  at  %^^  a  cental. 

30.  If  4  acres  of  land  cost  $321,  what  are  11^^  acres 
worth  ? 


31.  Divide  1  by  47^ 


3- 


32.  Bought  35  yards  of  carpeting  at  $1y^-o  a  yard,  3  cur- 
tains at  $|-  each,  5  chairs  at  $|  each;  what  was  my  bill? 

33.  A  can  w^alk  a  mile  in  I  of  an  hour,  and  B  in  ^  of 
an  hour;  in  a  race  of  22  miles,  which  will  win,  and  by  how 
much  ? 

34.  A  and  B  can  do  a  piece  of  work  in  10  days,  A  and 
C  in  12  days,  and  B  and  C  in  15  days;  in  w'hat  time  can 
the  three  working  together  do  it  ?  In  w^hat  time  can  each 
do  it  working  alone? 

35.  A  man  has  49|  acres  of  land;  he  sold  all  but  9|  acres 
of  land  for  $3190;  how  much  did  he  get  an  acre? 

36.  When  33^  yards  of  cloth  cost  $20,  what  is  the  price 
per  yard  ? 

37.  At  $2i  a  yard,  how  much  cloth  will  $^  buy? 

38.  How  many  sheep  must  I  sell  at  $3^  to  get  $169? 

39.  A  lady  divided  $3-|  among  some  poor  children,  giving 
them  $y^Q-  each;  what  number  of  children  were  there? 

40.  If  YQ  of  an  acre  of  land  is  Avorth  $23|,  what  is  1  acre 
worth  ? 

41.  Bought  50  sacks  of  potatoes  for  $62|;  what  will  12 
sacks  cost  at  the  same  rate  ? 

42.  Paid  \  of  my  money  for  a  lounge,  and  y^  of  it  for  a 
stove,  when  I  had  $106  left;  what  had  I  at  first? 

43.  A's  money  is  I  of  B's,  and  together  they  have  $1728; 
what  has  each  ? 

44.  What  do  I  receive  by  selling  17-|  bales  of  cotton,  each 
containing  b\  hundred  weight,  at  $18|  per  hundred  weight? 


100  CALIFORNIA    SERIES. 

45.  How  many  dipperfuls,  each  |  of  a  quart,  will  empty 
a  tub  containing  81^  quarts? 

46.  A  stockman  buys  a  certain  number  of  cattle  at  $24-| 
each  for  $588,  and  sells  them  at  $27|;  what  does  he  gain 
on  each  and  on  the  whole? 


47.  Divide  the  sum  of  3^  and  b-^-^  by 


4 
37' 


48.  A  has  \  as  much  money  as  B,  and  f  as  much  as  C; 
the  three  have  $2835;  what  has  each? 

49.  From  a  chest  of  tea  containing  63|  pounds,  If  was 
sold  for  $34;  Avhat  price  per  pound  was  obtained? 

50.  B's  money  is  If  times  A's,  and  C's  is  If  times  B's; 
all  together  have  $15300;  what  has  each? 

51.  If  I  of  a  cord  of  wood  costs  $5^,  what  will  \1\  cords 
cost  ? 

52.  I  buy  35f*6-  acres  of  land  at  one  time,  47f  acres  at 
another,  and  17f  acres  at  a  third;  I  sell  it  all  at  an  average 
rate  of  $40  an  acre;  what  do  I  receive  for  the  whole? 

53.  At  $1^  a  cental,  how  many  centals  of  wheat  can  be 
bought  for  $1000? 

54.  Find  the  sum,  difference,  and  product  of  |-|  and  |-|. 

55.  I  exchanged  5^  rolls  of  butter,  worth  40  cents  a  roll, 
and  10^  dozen  eggs,  worth  18  cents  a  dozen,  for  sugar  worth 
?!  cents  a  pound ;  how  many  pounds  of  sugar  did  I  receive  ? 

56.  Gained  $3|-  by  selling  12^  yards  of  cloth  for  $41fV; 
what  was  the  cost  per  yard  ? 

57.  8^  tons  of  Wellington  coal  at  $15  per  ton,  and  9f 
cords  of  wood  at  $7-i  a  cord,  amount  to  what? 

58.  Bought  40  acres  of  land  at  $63  an  acre.  Sold  -5^  at 
$72  an  acre,  -f^  at  $59^  an  acre,  and  the  remainder  for  $2^^ 
more  per  acre  than  I  paid  for  it;  what  did  I  gain  on  the 
whole  ? 

59.  f  of  189  is  what  fraction  of  567? 

60.  I  lend  A  a  certain  sum  and  B  twice  as  much.  -A 
pays  me  back  \  of  his  and  B  ^  of  his,  making  $150  received 
from  both;  what  did  I  lend  each? 


ARITHMETIC.  101 

61.  A  merchant  sold  35^  pieces  of  cloth,  each  piece  con- 
taining 47f  yards;  how  many  yards  did  he  sell? 

62.  Bought  40  bales  of  hay,  averaging  2^-^  hundred 
weight  a  bale;  how  many  hundred  weight  were  there? 

63.  How  many  tons  in  the  preceding  example,  at  20  hun- 
dred weight  to  a  ton  ? 

64.  There  is  an  average  of  365:j  days  in  a  year;  how 
many  hours? 

65.  Bought  14  cows  at  $23-|  a  head,  11  horses  at  $85 f  a 
head,  and  50  sheep  at  $2f  a  head;  what  had  I  left  from 
$1500? 

66.  Bought  15  sacks  of  potatoes  for  $12|,  and  sold  them 
for  -^  dollar  a  sack  more  than  I  paid;  what  did  I  receive 
for  them,  and  how  much  more  than  I  paid? 

67.  The  distance  by  rail  from  San  Francisco  to  Los 
Angeles  is  Off  times  the  distance  from  San  Francisco  to 
San  Jose.  The  sum  of  the  distances  is  533  miles;  what  is 
each  distance? 

68.  I  pay  3  men  $12.30  for  doing  a  piece  of  work.  The 
second  works  3  times  as  long  as  the  first,  and  the  third  \ 
as  long  as  the  first  and  second  together;  if  they  are  paid 
the  same  rate  per  day  what  should  each  receive? 

69.  Two  men  starting  at  the  same  point  travel  in  opposite 
directions  for  13|  hours.  One  travels  3^  miles  and  the 
other  3|  miles  per  hour:  how  far  apart  are  they  at  the  end 
of  the  time  ?     Draw  a  diagram  to  show  it. 

70.  Suppose  the  men  in  the  preceding  example  traveled 
in  the  same  direction,  how  far  apart  would  they  be  ?  Draw 
diagram. 

71.  Allowing  225|  pounds  to  a  barrel,  how  many  barrels 
of  sugar  will  2483^  pounds  make  ? 

72.  How  many  collars  at  $|  each  can  I  get  for  $2^? 

73.  What  is  the  average  value  of  4  horses  worth  $81^, 
$984,  $1051,  and  $112|,  respectively? 

74.  A  man  bought  35  watches  for  $15^  apiece  and  sold 


102  CALIFORNIA   SERIES. 

them  so  as  to  gain  $17^  on  the  whole;  what  did  he  get 
apiece  for  them  ? 

75.  A  boy  has  375  oranges.  He  sells  y\  of  them  at  one 
time  and  y\  at  another;  what  are  the  remainder  worth  at 
If  cents  each? 

76.  A  merchant  sold  two  pieces  of  cloth  for  $241^.  One 
piece  contained  30^  yards,  the  other  42^  yards;  what  aver- 
age price  per  yard  did  he  get? 

77.  Richard -^can  walk  ^  as  fast  as  Walter.  In  a  certain 
time  both  together  walked  5|-  miles;  what  part  of  the  dis- 
tance did  each  walk? 

78.  At  $1  a  day  what  will  a  man  earn  in  1  year,  leaving 
out  60  days  for  Sm:»days  and  holidays? 

79.  If  11^  boxes  of  oranges  cost  $28|,  how  many  boxes 
can  I  get  for  $22^? 

80.  Fred  worked  2|  times  as  long  as  Frank  at  |  as  much 
per  day.     They  received  $24^;  what  part  should  each  have  ? 


DECIMAL  FEAOTIONS. 

If  you  divide  a  unit,  or  1,  into  10  equal  parts,  what  is 
each  part  called?  How  many  tenths  make  a  unit?  If 
you  divide  each  of  the  lOths  into  10  equal  parts,  how  many 
pieces  will  there  be  and  what  is  each  called?  How  many 
hundredths  in  1  tenth?  If  you  divide  each  hundredth 
into  10  equal  parts,  how  many  pieces  will  there  be  and  what 
is  each  called  ?     How  many  thousandths  in  1  hundredth  ? 

Review  Obs.  on  p.  6. 

What  is  the  first  figure  on  the  left  of  the  decimal  point 
called?  The  second?  The  third?  The  fourth?  How 
many  units  make  1  ten?  How  many  tens  make  1  hun- 
dred?   What  part  of  100  is  10?     Of  10  is  1  ? 

Since  numbers  decrease  by  10  fold  from  left  to  right  we 
may  go  on  in  our  decimal  notation  beyond  the  decimal 


ARITHMETIC.  103 

point  on  the  right,  and  make  the  name  of  each  succeeding 
place  Y^o"  t^^  vakie  of  the  preceding.  -Thus,  starting  with 
units,  the  first  figure  on  the  right  of  the  point  will  be  y\j-  of 
units  or  lOths.  5.7  is  5  and  ^^j-.  What  will  the  second 
figure  on  the  right  of  the  point  be  called  ?  The  third  ?  The 
fourth?  How  do  we  mark  the  absence  of  number  in  any- 
decimal  place?     (See  Obs.,  p.  6.) 

Fractions,  then,  whose  denominators  are  10,  100,  1000, 
etc.,  may  be  written  decimally ;  the  denominator  being  indi- 
cated by  the  number  of  places  on  the  right  of  the  decimal 
point,  and  not  expressed,  as  in  the  common  form. 

Put  a  diagram  similar  to  the  following  on  your  slate, 
extending  it  further  if  necessary;  under  it  place  the  mixed 
numbers  and  fractions  of  Exercise  143  in  their  proper 
places,  and  practice  reading.  In  reading,  use  "  and  "  at  the 
decimal  point  only. 

Read  the  fractional  part  as  a  whole  number  first,  and  add 
the  decimal  name  of  the  last  figure;  thus, 

32575  and  4763  hundred-thousandths. 


QQ 

r^ 

■+3 

^ 

. 

a 

, 

c3 

w. 

n5 

o 

§ 

OQ 

. 

QQ 

c5 

-1-3 

OQ 

c3 

OQ 

-1-3 

r^-' 

m 

:^ 

-C5 

^ 

c3 

»— ' 

^ 

o 

(V 

CO 

O 

g 

^3 

^ 

. 

m 

X 

'T^ 

w. 

-^ 

'^ 

cc 

~t-^ 

-(-3 

C 
^ 

"P 

1 

/-( 

O 

a; 

.1— 1 

C 

o 

O 

r2 

3 

-♦-i 

^ 

l-M 

4-3 

P 

-*-^ 

r-—t 

-*^ 

•h^ 

r^ 

\ 


3257    5. 0   4763 

'  The  number  of  figures  on  the  right  of  the  point 
Observe,  i       corresponds  to  the  number  of  O^s  in  the  denom- 
y      inator  of  the  fraction. 

EXERCISE   143. 

1.  75.14  3.  131.131  5.     7.007  7.  1389.9 

2.  .125  4.       .0785  6.  .13147  8.     .0091 


104  CALIFORNIA   SERIES. 


9. 

857.14 

15. 

2.0404 

21. 

480.7 

27. 

3150.071 

10. 

85.0714 

16. 

7814.002 

22. 

526.114 

28. 

4090.07 

11. 

.07408 

17. 

7.0707 

23. 

.070107 

29. 

293.0293 

12. 

.00291 

18. 

20.0003 

24. 

.1410 

30. 

47.141 

13.  405.01        19.  171.4112       25.  82.1073       31.       29.641 

14.  78.78       20.  27141.75       26.  1.01010       32.  10.1 

EXERCISE    144. 

Write  each  example  of  Exercise  143  with  denominators, 
thus,  7r  1  4 

Write  each  in  words,  also;  thus, 

Seventy-five  and  Jourteen  hundredths. 

EXERCISE   145. 

AVrite  the  following  in  decimal  and  in  common  forms, 
and  reduce  the  fractional  parts  in  the  common  form  to 
lowest  terms: 

1.  Twenty-five  and  twenty-five  hundredths,  9  and  114 
thousandths,  7  and  5  tenths,  11  and  8  thousandths. 

2.  74  and  99  ten  thousandths,  11  and  45  hundred  thou- 
sandths, 4  thousandths,  4  hundredths. 

3.  75  hundredths,  75  ten  thousandths,  40  and  40  hun- 
dredths, 4  thousand  and  4  thousandths. 

4.  91  hundredths,  91  tenths,  400  thousandths,  121  hun- 
dredths. 

5.  90  tenths,  57  thousandths,  5  and  11  thousandths,  72 
and  6  tenths. 

6.  87  and  54  hundredths,  90  and  8  tenths,  117  and  41 
thousandths,  25  and  9  thousandths. 

7.  238  and  12  thousandths,  171  and  125  thousandths, 
328  and  10  thousandths,  190  and  8  thousandths. 

8.  2  tenths,  24  tenths,  120  tenths,  175  tenths. 

9.  830  hundredths,  375  hundredths,  57  hundredths,  9 
hundredths. 

10.  2496  thousandths,  7125  thousandths,  125  ten  thou- 
sandths, 25  thousandths. 


ARITHMETIC.  105 

EXERCISE   146. 

(1)  Write  10  mixed  numbers  or  fractions  of  your  own  in 
the  common  form,  using  10,  100,  1000,  etc.,  for  denomina- 
tors; (2)  the  same  in  decimal  form;  (3)  the  same  in  words; 
(4)  write  the  first  examples  with  the  fractions  in  their  low- 
est terms.     Bring  to  the  class  for  dictation. 

EXERCISE   147. 

Write  the  following  in  common  form  and  reduce  to  low- 
est terms: 

1.  .25,  .75,  .125.  6.  .144,  .0256,  .075. 

2.  .375,  .088,  .048.  7.  .84,  .164,  .175. 

3.  .0175,  .35,  .015.  8.  .8,  .50,  .0625. 

4.  .16,  .016,  1.75.  9.  .1875,  .625,  .18. 

5.  .33^,  .661,  .78.  10.  .95,  .3125,  .105. 


DOLLARS  AND  CENTS  WRITTEN  DECIMALLY. 

The  decimal  notation  is  employed  in  writing  dollars  and 
cents  in  United  States  money.  There  are  100  cents  in  1 
dollar.  Hence  any  number  of  cents  are  so  many,  hun- 
dredths of  a  dollar;  thus, 

5  cents  are  yfo-,  or  .05  of  a  dollar;  20  cents  -f^Q,  or  .20. 

12  dollars  6  cents  is  written  $12.06;  17  dollars  37  cents, 
$17.37;  18  dollars  12^  cents,  $18.12^  All  rules  for  opera- 
tions in  decimals  are  equally  true  of  United  States  money. 

Observe. — The  decimal  point  is  'placed  at  the  right  of  dollars. 

EXERCISE    148. 
Read  the  following  as  dollars  and  cents: 

1.  $7.02  5.  $175.75  9.  $708.09  13.  $927.06 

2.  $25.50  6.  $38.25  10.  $150.12^  14.  $41 .62^ 

3.  $137.37-1  7.  $450.80  11.  $45.33^  15.  $108.03 

4.  $98.01  8.  $92.90  12.  $128.07  16.  $29.66f 


106  CALIFORNIA   SERIES. 

EXERCISE  149. 

Write  20  numbers  of  your  own,  representing  dollars  and 
cents  decimally,  and  bring  to  the  class  for  reading  and  dic- 
tation. 

To  change  any  fraction  from  the  common  to  the  decimal 
form. 

How  is  a  fraction  reduced  to  higher  terms?  (See  pp.  75 
and  76.)  How  do  you  change  ^  to  a  fraction  having  a  de- 
nominator 10?  \  is  how  many  lOths?  f  are  how  many 
lOths?  f?  f?  i  is  how  many  lOOths?  |?  i  is  how 
manylOOOths?     f?     |?     |? 

Write  each  result  in  decimal  form. 

We  see  from  this,  that,  to  change  any  fraction  from  the 
common  to  the  decimal  form,  we  reduce  it  to  a  fraction 
having  10,  100,  1000,  etc.,  for  a  denominator,  and  write 
decimally. 

EXERCISE  150.    (Oral.) 

Change  to  decimal  denominator  and  express  decimally: 


13       4       7 
■■■•     1  0?    5?   2  0- 

4. 

2      13      19 
5'    2  05    2  5- 

7. 

2      19      24 
45    505    25- 

9        4       1111 
^'     2" 5"7    5"0"?  "2¥- 

5. 

17      3     18 
"2  0"?    4  5    2  5' 

8. 

4  5       1        1 
505    205    5- 

q      11       9       27 
^'     2  55    5  0?   -2  5- 

6. 

49      21      17 
505    2  55    5  0* 

9. 

7       14      37 
255    205    50 

Again:  5^|-= 

= 

5  0- 
1  0^ 

—  500 
~1  00 

zr^: 

fU^=5.000 

1^^  of  5,  or 

1 

8 

of 

5  0  0  0 
1000 

t¥^V=8)  5.000 

.625 
Therefore  |=.625. 

Hence,  to  change  common  to  decimal  fractions,  annex 
ciphers  to  the  numerator  and  divide  by  the  denominator. 

If  the  denominator  contains  no  other  prime  factors  than 
2  or  5,  the  division  will  be  exact;  and  the  number  of  places 
will  be  equal  to  the  largest  number  of  times  2  or  5  is  con- 
tained as  a  factor. 

Thus,  in  yf-,  the  division  is  exact,  because  5  is  the  only 
prime  factor  in  125,  and  there  will  be  3  places,  because  125 


ARITHMETIC.  107 

contains  5^;  in  -^^  the  division  is  not  exact,  because  375 
has  the  factor  3.  When  the  division  is  not  exact,  carry  it 
to  3  or  4  places  and  express  the  remainder  in  the  form  of  a 
common  fraction. 

EXERCISE   151.    (Oral.) 

Tell  by  inspection  whether  the  following  are  exact  deci- 
mals; and,  if  exact,  how  many  places  in  the  decimal: 


2.S 
40 

1  9 

2  0 

4 
125 

1  .3 
5  0 

1  9 
75 

40 
60 

17 
150 

13 
105 

1  8  1 
125 

.3  6 
17  5 

1  9 

2  00 

4  3 

10  0 

78 
2  7  5 

_3JL 
300 

17 
250 

1  1 
¥T5" 

In  performing  work  where  denominators  are  as  small  as 
in  the  preceding  exercise,  it  is  better  to  Avork  mentally, 
reducing  as  voii  go;  thus, 

^=.5^.57\=.575 ;  ^=.0-.S=.00^S=.002i. 

Work  the  preceding  exercise  in  this  way. 

EXERCISE   152. 

Change  the  fractions  in  Exercise  150  to  decimals  by  this 
method. 

Also  the  fractional  part  in  Exercises  97  and  98,  and  re- 
write the  mixed  numbers  in  decimal  form. 

EXERCISE    153.  ,  . 

Write  10  common  fractions  of  your  own,  and -change  to 
decimals.     Bring  to  the  class  for  dictation. 


CIRCULATING  DECIMALS. 

In  cases  where  the  division  in  the  preceding  work  is  not 
exact,  the  figures  of  the  quotient  will  begin  to  repeat  at 
some  point  of  the  division,  producing  what  is  called  a  circu- 
lating decimal.  The  circulate,  or  repeating  part,  is  marked 
by  a  dot  over  the  first  and  last  of  the  repeating  figures. 

The  number  of  places  before  the  circulate  begins  will 
equal  the  greatest  number  of  2's  or  5's  in  the  denominator. 


108  CALIFORNIA   SERIES. 

When  special  accuracy  is  required,  however,  it  is  better 
to  divide  until  the  2's  and  5's  are  all  canceled,  and  express 
the  remainder  as  a  common  fraction.  Thus,  ■2V-o=.4409, 
expressed  as  a  circulate;  or  .44jiy,  expressed  fractionally. 


ADDITION  AND  SUBTRACTION  OF  DECIMALS. 

Decimals  containing  fractions  are  written  and  worked, 
for  adding  and  subtracting,  like  whole  numbers.  How 
must  they  be  written,  and  why?  (See  explanations,  pp.  18 
and  24,  Addition  and  Subtraction.) 

EXERCISE    154. 

AVrite  properly  and  add  the  numbers  in  examples  1  to  8, 
Exercise  143;  same  with  9  to  16,  17  to  24,  25  to  32.  Also 
find  the  sum  of  the  numbers  in  each  row.  Add  the  num- 
bers in  each  example  of  Exercise  145. 

EXERCISE    155. 

Find  the  difference  between  Example  1,  Exercise  143, 
and  each  remaining  example  in  the  same  column;  Exam- 
ple 9  and  each  remaining  example  in  the  same  column; 
Example  17  and  each  remaining  example  in  the  same  col- 
umn; Example  25  and  each  remaining  example  in  the  same 
column. 

EXERCISE  156. 

Add  the  numbers  in  each  example.  Exercise  97,  as  they 
are;  change  to  decimals  and  add;  compare  results. 
Subtract  in  the  same  way. 

EXERCISE    157. 

Change  the  common  fractions  to  decimals,  and  perform 
examples  1,  2,  3,  8,  10,  17,  and  20,  Exercise  114. 


ARITHMETIC.  109 

MULTIPLICATION  OF  DECIMALS. 

Multiply  3.728  by  .18. 

OPERATION.      Explanation. — Any  number  of  units  times  a  certain 

3 .  /  2  8     denomination  gives   tliat   denomination  as  a  product. 

.18      Hence,  18X3.728=67.104;  but  the  multiplier  is  ^'h  or 

9  <m  9  /t     ^''^  ^^  ^^'     Tlierefore  the  product  will  be  too  of  67.104 

or  .67104.     Whence  the  law  for  multiplying  decimals : 

O  y  r)  Q  i-  «       o 

Point  off  from  the  right  as  many  places  in  the  product 


.6  7104     as  there  are  in  both  'multiplicand  and  multiplier. 

EXERCISE    158. 

Use  numbers  in  examples  1  to  8,  Exercise  143,  as  multi- 
plicands, and  75.14  for  a  multiplier  of  each:  also,  .125  as  a 
multiplier;  9  to  16  as  multiplicands,  and  examples  3  and 
4  as  multipliers;  17  to  24  as  multiplicands,  and  5  and  6  as 
multipliers;  25  to  32  as  multiplicands,  and  7  and  8  as  mul- 
tipliers. 

Finish  with  one  multiplier  before  using  another. 

EXERCISE  159. 
Multiply  the  numbers  in  each  example,  Exercise  97,  as 
they  are;  change  to  decimals  and  multiply.     Compjjre  re- 
sults.    Perform  the  work  of  Exercise  119,  changing  com- 
mon fractions  to  decimals. 


EXERCISE 
Find: 

160 

.     (Analysis,  General  Fori 

n  G,  p. 

93.) 

1. 

.06  of  725. 

8. 

.33^  of  515.1. 

15. 

.8  of  3.55. 

2. 

.8  of  42.5. 

9. 

.05  of  480. 

16. 

.025  of  96. 

3. 

.125  of  7.84. 

10. 

.175  of  .764. 

17. 

.28  of  250. 

4. 

.03  of  17.28. 

11. 

.04  of  57.75. 

18. 

1.05  of  1400 

5. 

.12^  of  4.096. 

12. 

.9  of  1.044. 

19. 

1.2  of  380. 

6. 

.161  of  256. 

13. 

.06:1  of  72400. 

20. 

.45  of  920. 

7. 

.25  of  2.444. 

14. 

.15  of  245.4. 

Also  perform  the  above  by  changing  the  multiplier  in 
each  example  to  a  common  fraction.     Compare  work. 


110  CALIFORNIA   SERIES. 

EXERCISE   161. 

Perform  Exercises  123  and  128,  changing  the  fractions 
to  decimal  form. 


DIVISION  OF  DECIMALS. 

Di^dde  .67104  by  3.728. 

OPERATION.  Explanation. — Any    denomination    di- 

3.728).67104(.18      vided  by  tlie  same  denomination  gives  miits 

3  7  '2  8  ^^^  '^  quotient;    .671-^3.728  gives  0  miits 

9  Q  k  9 1  ^^^  '^  quotient ;  the  remaining  places  in  the 

9  Q  K  9  4  dividend  will  be  the  number  of  places  to 

point    off    in    the    quotient    giving    .18. 

Whence  the  law  for  division  of  decimals : 

Point  ojf  as  many  places  from  the  right  in  the  quotient  as  those  in 
the  dividend  exceed  those  in  the  divisor. 

When  the  division  is  not  exact,  carry  the  answer  to  3  or  4  places 
beyond  the  point;  in  money  operations,  to  2  places,  adding  1  to  the 
second  figure  if  the  third  figure  should  be  5  or  more. 

Where  special  accuracy  is  required,  express  the  remainder  as  a 
common  fraction. 

In  actual  practice  it  is  a  shorter  and  surer  way  to  draw  a  vertical 
line  in  the  dividend  after  the  figure  whose  denomination  is  that  of 
the  divisor,  first  annexing  ciphers  to  the  dividend  if  necessary: 
when  the  division  has  reached  this  line  put  a  point  in  the  quotient. 

Divide  44.232  by  .12. 

OPERATION. 

"19W4   9319         Draw  a  vertical  line  after  hundredths,  that 
— — ^^  Q    ^     being  the  denomination  of  the  divisor.    The  part 
o  b  8 .  b     ^j  ^YiQ  dividend  on  the  left  of  the  hne  contains 
the  divisor  368  times.     Then  comes  the  point. 

EXERCISE   162. 

Divide  examples  1  to  8,  Exercise  143,  by  .02  (short 
division) ;  examples  9  to  16  by  .095;  17  to  24  by  327;  25  to 
32  by  1.01. 


:frc. 


ARITHMETIC.  Ill 

EXERCISE   163.  ^^ 

Divide  each  number  in  this  row  by  25: 
1.  3.  4.  7.  8.  1.1  7.8  .001 

Divisor  .1  for  this  row: 
10.  .001  75.  7.5  .75  .1  100. 

Divisor  150  for  this  row: 
45.  .0450  .75  7.5  3.  10.  15. 

EXERCISE   164. 

Divide,  in  Exercise  97,  the  greater  number  by  the  less, 
using  the  numbers  as  they  are;  change  to  decimals  and 
divide.     Compare  results. 

EXERCISE   165. 
Perform  the  work  of  Exercises  132  and  135  by  decimals. 

EXERCISE    1  66.     (Analysis,  Geueral  Form  H,  p.  93.) 

9.  120  is    .15  of  what? 


1. 

75  is 

.03  of 

what? 

2. 

125  is 

.05  " 

a      9 

3. 

128  is 

.2  " 

u      9 

4. 

296  is 

.04  '^ 

u      9 

5. 

144  is 

.12  " 

a      9 

6. 

50  is 

.025  " 

u      9 

7. 

150  is 

.1  " 

a      9 

8. 

99  is 

.011  " 

u      9 

10. 

240  is 

1.20  '' 

u 

? 

11. 

196  is 

1.4     '' 

u 

9 

12. 

28  is 

.07  " 

a 

? 

13. 

13  is 

.08  " 

a 

? 

14. 

14  is 

.09  '' 

sU 

? 

15. 

15  is 

.10  " 

a 

? 

16. 

16  is 

.11  " 

u 

9 

CONTRACTED  MULTIPLICATION  OF  DECIMALS. 

In  multiplying  decimals  having  several  places  on  the 
right  of  the  point,  where  the  product  is  desired  only  to  2  or 
3  places,  the  work  may  be  greatly  shortened  by  multiply- 
ing only  those  denominations  that  produce  the  required 
places.     Thus, 

Multiply  428.9543  by  17.454:    2  decimal  places  required. 


112  CALIFORNIA   SERIES. 

OPERATION.  Explanation. — Place  denominations  of  the  same 

4  2  8.9  543     name  under  each  other.     For  convenience,  use  the 

1  7.454        ^^^^  figure  of  the  multipher  first,  if  there  is  a  unit 

q  r\  Q  9  no  figure,  since  units  times  any  denomination  gives 

Q  Q  qV  ^  tl^^^t  denomination.     Begin  with  hundredths  in  the 

multipUcand,   the  required  denomination  in  the 

l/l.Oo  answer.     7X5  hundredths^35  hundredths+3  hun- 

21.45  dredths  (7X4  thousandths=28  thousandtlis,  w^hich 

1.7  2  being  2H  oi"  more  we  call  3)==38  hundredths;  and 

748 6~97  ^^  ^^^*     ^^^  multiplying  by  1  ten  we  go  one  place 

farther  to  the  right  in  the  multiplicand;  by  4  tenths, 

one  place  farther  to  the  left;  and  so  on. 

This  operation  saves  so  much  labor  that  it  should  be  used,  when 
available,  throughout  the  work. 

EXERCISE  167.    (Written.) 

Perform  by  contracted  multiplication;  the  first  5,  to  2 
places,  the  remainder,  to  3  places. 

1.  H17.87X.0783.  6.  85.0714x131.131. 

2.  191.45X.173.  7.  7814. 002X. 0785. 

3.  .956X1.413.  8.  20.0003x7.007. 

4.  .7854X3.1416.  9.  .13147x480.7. 

5.  91.4726X7.141.  10.  171.4112X293.0293. 


CONTRACTED  DIVISION  OF  DECIMALS. 
Divide  4129.7854  by  47.62143;  2  decimal  places  required. 

OPERATION. 

47021143)  41  2^9.71854(86.72       Explanation.- The    con- 
^•^007  traction  consists  in  omitting 

a  figure  from  the  right  of  the 
divisor,  instead  of  bringing 
down  one  at  the  right  of  the 
dividend,  in  each  successive 
division .  The  last  divisor  is 
the  left-hand  figure  of  the 
divisor.  Now  take  the  divi- 
dend to  include  the  same 
denomination  as  the  highest 


3201 

2857 

344 
333 

11 
10 

PROOF. 

47.62143X867 

2 

reserving 

1  dec 

Pl 

47.62143 

86.72 

285.7 

38097 

333 

10 

ARITHMETIC.  113 


place  in  the  divisor,  and  the 
quotient  will  he  units  (see 
p.  110);  therefore  go  to  the 
right  of  this  denomination  as 
many  places  as  you  wish  to 
reserve  decimal  places  in  the 
quotient. 

Cut  down  the  divisor  from 
the  right  until  it  is  contained 
in  this  dividend. 


4129.7 

1.  3.4268731 --.284638413,  reserving  2  decimal  places 

2.  .04278593— .02872539,  reserving  3  decimal  places. 


PRACTICAL  AVORK  IN  DECIMALS. 

EXERCISE   168.   (Written.) 

1.  In  1880  California  raised  240.25  acres  of  cotton,  aver- 
aging .63  of  a  bale  to  the  acre;  how  many  bales  were  pro- 
duced ? 

2.  If  there  are  475  pounds  of  cotton  in  a  bale,  what  num- 
ber of  pounds  to  the  acre  was  produced  ? 

3.  At  $40  an  acre,  what  are  3  fields  worth,  containing, 
respectively,  17.6  acres,  23.25  acres,  and  42.625  acres? 

4.  A  man  owning  .3125  of  a  ship,  sold  .2  of  his  share; 
what  part  had  he  left? 

5.  What  are  36  dozen  eggs  worth  at  $.12^  per  dozen? 

6.  A  man  divided  his  ranch  of  648.96  acres  into  8  equal 
fields;  how  many  acres  did  each  field  contain? 

7.  At  $2.25  each,  how  many  books  can  you  buy  for  $27? 

8.  How  far  will  a  horse  travel  in  11  hours  at  the  rate  of 
6.75  miles  an  hour? 

9.  A  bushel  contains  2150.42  cu.  in.;  5.16f  bushels  con- 
tain how  many  cu.  in.? 

10.  A  man  earns  $1.37|  a  day;  if  he  works  296  days  dur- 
ing the  year,  what  will  he  earn? 

8— A 


114  CALIFORNIA   SERIES. 

11.  There  are  16.5  feet  in  1  rod;  how  many  rods  are  in 
272.25  feet? 

12.  A  man  walks  32.75  miles  on  Monday,  29.8  on  Tues- 
day, 27.41  on  Wednesday,  40.5  on  Thursday,  ol.G6|  on  Fri- 
day, and  25.33:^  on  Saturday;  how  far  did  he  walk  during 
the  week? 

13.  What  was  his  average  distance  per  day? 

14.  A  real  estate  agent  having  3218  acres  of  land  to  sell, 
sold,  on  different  occasions,  278.15  acres,  392.14  acres,  171.9 
acres,  429.51  acres,  and  530.875  acres;  what  liad  he  left? 

15.  Allowing  2.625  yards  to  a  pair,  how  many  pairs  of 
pants  can  be  made  from  a  piece  of  cloth  containing  42 
yards  ? 

16.  There  are  231  cubic  inches  in  a  gallon,  and  31.5  gal- 
lons in  a  barrel;  how  many  cubic  inches  are  in  a  barrel? 

17.  How  many  rods  of  fence  will  surround  a  field  32.0625 
rods  long  and  28.4375  rods  wide? 

l&j-  How  many  turns  will  the  driving-wheel  of  a  locomo- 
tive make  in  going  1  mile,  the  wheel  being  21.96  feet  in 
circumference? 

19..  I  bought  12  horses  at  $81,875  apiece,  and  gave  a 
1000-dollar  note  in  pajniient;  what  change  did  I  receive? 

20.  At  $7.75  per  cord,  how  many  cords  of  wood  can  be 
bought  for  $162.75? 

21.  I  have  4  fields;  the  first  contains  7.231  acres,  the  sec- 
ond 9.124  acres,  the  third  6.715  acres,  and  the  fourth  i  as 
much  as  the  other  3  together;  what  do  they  all  contain? 

22.  What  are  all  worth  at  $50  an  acre? 

23.  I  spend  .08  of  my  money  one  day,  .16  a  second  day, 
i  of  it  a  third;  if  I  have  $26  left,  how  much  had  I  at  first? 

24.  A  man  on  a  journey  goes  ^  of  it  on  Monday,  and  .450 
on  Tuesday;  how  much  has  he  left  to  perform? 

25.  I  have  15J  cords  of  wood  in  one  pile,  17.66|  in  a  sec- 
ond, 14^  in  a  third,  and  15.125  in  a  fourth;  how  many 
cords  in  all? 


ARITHMETIC.  115 

26.  How  much  is  it  worth  at  $7f  a  cord? 

27.  The  distance  around  a  pond  is  .59375  of  a  mile;  how 
many  times  around  it  can  I  travel  in  8  hours,  traveling  4f 
miles  per  hour? 

28.  How  many  fence  rails  l.b  feet  long  will  go  3  times 
around  a  field  ISyV  rods  long  and  10.1875  rods  wide  ?  Draw 
diagram. 

29.  The  rainfall  at  Sacramento  for  the  year  ending  with 
August,  1880,  was  26.744  in.;  1881,  26.134  in.;  1882, 
16.283  in.;  1883,  18.3  in.;  1884,  24.78  in.  What  was  the 
average  yearly  rainfall  for  that  time? 

30.  We  inhale  about  2.125  gallons  of  air  every  minute; 
how  much  do  we  inhale  in  an  hour? 

V   31.  Sold  17.125  tons  of  hay  at  $9f  per  ton;   what  was 
received  for  the  hay? 

32.  Two  men  start  from  the  same  place  at  the  same  time 
and  travel  in  opposite  directions;  one  goes  4.64  miles  an 
hour,  the  other  5.16;  how  far  apart  are  they  in  13  hours? 

33.  When  they  are  107.8  miles  apart,  how  many  l^ours 
have  they  traveled? 

34.  The  distance  around  a  circle  is  3.1416  times  the  dis- 
tance across  it.  If  I  can  walk  across  a  circular  field  in  48 
seconds,  how  long  will  it  take  me  to  walk  around  it? 

35.  If  the  distance  through  the  earth  is  8000  miles  what 
is  the  distance  around  it? 

Perform  decimally  examples  1,  2,  3,  5,  8,  10,  14,  17,  21, 
25,  27,  30,  35,  37,  38,  39,  40,  52,  53,  57,  65,  71,  73,  76, 
Exercise  142. 


SHOUT  METHODS  IN  MULTIPLICATION. 

An  aliquot  part  of  a  number  is  such  a  number,  whole  or 
fractional,  as  is  contained  in  it  an  exact  number  of  times. 
Thus,  20,  25,  and  33^  are  aliquot  parts  of  100,  being  con- 
tained respectively  5,  4,  and  3  times  in  100. 


116  CALIFORNIA   SERIES. 

1.  To  multiply  by  an  aliquot  part  of  100,  1000,  etc. 

Multiply  447  by  33^ 
OPERATION.        Explanation. — Multiplying  by  100  gives  a  product 
3)44700    3  times  too  large,  since   100=3X33i;    dividing  by  3 
1  4900    gi'^^s  ^^6  ^^'^^  product. 


2.  To  multiply  by  9,  99,  999,  9999,  etc. 

Multiply  5728  by  99.  Multiply  387  by  999. 

OPERATION,  OPERATION. 

572800.=100X5728    ,  387000. 

5728.=^      1X5728 387^ 

5(37072.^   99X5728  386613. 

3.  To  multiply  when  a  part  of  the  multiplier  is  a  multi- 
ple of  another  part. 

Multiply  3216  by  357. 

OPERATION. 

3216 

OCT 

"^"^  '  Explanation. — Multiply  by  7  for  the  first 

22512  product;    then  this  product  by  5,  since  5X7 

1  12  5  60  times  a  number  is  35  times  that  number. 


1148112 


4.  To  multiply  numbers  whose  tens  are  alike  and  the 
sum  of  whose  units  is  10. 

43X47=(50X40)  +  {3X7)=2021. 
43 

■47 

^ 7v^  Explanation. — The    product    of 

the  tens  by  1  more  than  itself  gives 

"^  "  ^^^^     /  X  4  U  \  hundreds ;  and  the  product  of  the 

120-   3X40  -  =50X40.  units,  units. 
1600=40X40  ) 
2021 

The  preceding  method  may  be  applied  to  mixed  num- 


ARITHMETIC.  117 

bers  whose  integral  parts  are  alike  and  the  sum  of  whose 
fractional  parts  is  1. 

5|-X54=(6x5)  +  aXf)=30i|. 

5.  To  find  the  product  of  two  numbers  whose  mean 
number  is  easily  squared. 

57x63=3600—9=3591. 

Here  the  mean  or  middle  number  is  60,  it  being  3 
greater  than  57,  and  3  less  than  63. 

The  result  is  60'^— 3^=3591. 

The  same  process  may  be  applied  to  mixed  numbers. 
4|x5i=5^-i^=24||. 

6.  To  multiply  a  number  by  itself  or  square  it. 

64^=60^+2x4x60+4^=4096. 
64 
64 


1  6=4"  Explanation. — Square  the  tens ;  add  the  prod- 

2  4Q__4v/gQ  net  of  the  tens  by  twice  the  units,  and  the  square 
240=4X60   ^f  the  mats. 

3  600=60^ 

4096 

The  same  result  may  be  reached  by  reversing  the  process 
in  No.  5. 

64^=60  X  68+ 4-=4096. 

EXERCISE   169.    (Written.) 

1.  4721X999.  9.  82x78.  17.  41x39. 

2.  117X113.  10.  104X106.  18.  lUXl2i 

3.  576x33i  11.  4184X125.  19.  99x4201. 

4.  875X328.  12.  396xl6§.  20.  68x62. 

5.  llfXllf.  13.  3248Xl2i  21.  75x2320. 

6.  7^X8|.  14.  81X89.  22.  166fX891. 

7.  2160X99.  15.  1064x25.  23.  8iX8f 

8.  41X41.  16.  94X94.  24.  72xT2. 


118  CALIFORNIA   SERIES. 

SHORT  METHODS  IN  DIVISION. 

1.  To  divide  by  an  aliquot  part  of  100,  1000,  etc. 

Divide  4256  by  33f 

OPERATION.  Explanation. — 100  is  contained  42 

3X42+1=127  Quotient.  times  with  56  Rem.     33^  (i  of  100) 

56-33^=221  Remainder.    ^^  contained  3x42+(56^33t)  or  127 

times  with  22|  Rem. 

Divide  4256  by  14f . 

7x42+3=297  Quo. 

56— (3Xl4|-)=13i-Rem. 

2.  To  divide  by  a  number  a  little  less  than  100,  1000,  etc. 

Divide  31241  by  99. 

OPERATION.  Explanation. — 100    is    contained    312 

QQ\o-|2  41  times  with  a  remainder ;   99  is  contained 

-|  o  312   times   with   312   units   additional   re- 

mainder.    Dividing  this  remainder,  100  is 
contained  3  times  with  12  remainder,  so 


Qno.=3  15  5  6=Rem.  99  jg  contained  3  times  with  3  more  re- 
mainder. Tlie  sum  of  the  quotients  is  315,  and  the  sum  of  the 
remainders,  56.  When  the  sum  of  the  remainders  equals  or  ex- 
ceeds the  divisor,  it  must  be  again  divided  in  the  same  way. 

If  the  divisor  be  98  or  97  the  additional  remainder  is  the  quotient 
times  2  or  3. 


Divide  31241 

by 

998. 

998)31 

241 
6  2 

Quo. --3  1 130  3 

EXERCISE  170. 

5280--16|. 

5.  9825--125. 

28171-f-99. 

6.  7200--lli. 

7428^97. 

7.  41256- 

-997. 

:Eem. 


1.  5280--16|.  5.  9825--125.  9.  21047--98. 

2.  28171-f-99.  6.  7200--lli  10.  7519--166|. 

3.  7428^97.  7.  41256-f-997.  11.  2763--114f 

4.  7800--25.  8.  5386^333^.  ]2.  3672-=-212i. 


ARITHMETIC. 


119 


BILLS. 

When  one  person  sells  goods  to  another,  or  works  for 
another,  he  writes  to  the  buyer  or  employer  an  account  of 
the  things  sold  or  work  performed,  with  dates,  prices,  and 
amount.     Such  a  writing  is  called  a  Bill., 

When  the  bill  is  paid,  the  person  receiving  the  money 
signs  his  name,  with  the  words,  "  Received  Payment,"  or, 
"  Paid,'"  at  the  end.     This  is  called  receipting  the  bill. 

The  abbreviation  Dr.  for  debtor  (ower)  is  sometimes  used, 
showing  that  the  person  first  named  in  the  bill  owes  the 
money. 


Mk.  F.  E.  Adams, 


Los  Angeles,  ]\Iar.  26,  1886. 
Bought  of  Ellis,  Wells,  &  Co. 


1886 

Peb. 

10 

i  i 

i  i 

a 

15 

Mar. 

1 

i  I 

i  I 

i( 

10 

10  tt).  Gran.  Sugar,  @  O^i-f 

3   "     Cheese,     .  ''  15^ 

2  bags  Flour,       .  "  $1.25 

1       2  tt^.  Coffee,         .  '•  37>^'^ 

2  "    Tea,       .     .  "  bM 

3  rolls  Butter,      .  "  60^ 


$ 

ct. 

95 
45 

$ 

2 

50 
75 

1 

10 

1 

80 

7 

ct. 


-)0 


Rec'd  Payment, 

Ellis,  Wells,  &  Co. 


San  Jose,  Cal.,  Mar.  1,  1886. 
Mr.  a.  S.  Ames, 

To  J.  E.  Symoxds,  Dr. 

To  5  days' Labor,  @  $1.25 $6.25 

Rec'd  Payment, 

J.  E.  SVMONDS. 

EXERCISE    171. 

Copy,  on  paper  or  slate,  carry  out  the  items,  and  receipt 
the  bills  found  on  page  120. 


120 


CALIFORNIA   SERIES. 


Oakland,  June  1,  1886. 


Mr.  Geo.  H.  Jones, 


To  Barnes  &  Cole,  Dr 

• 

1886 

May 

30 

To    1  bbl.  Gran.  Sugar,  245   ft).   @  8^ 
"     1  10-pound  sack  Oatmeal,  .     .     . 
"     3  ft).  Honey,       .     .     .     @    12)^^ 
"     4  sacks  Flour,     .     .     .      "    $1.35 
"     3  ft).  Eaisins,      ...      "        15*/ 
*'     7  doz.  Eggs,  .     .     .     .      "        16<? 
"  10  ft).  Crackers,         .     .      "      8)^^ 
"     Icaddy  Japan  Tea,  22  ftj.,"        G5</' 
"     10- ft),  sack  Salt,      .     .      "      3K'? 

25 

2. 

Mercantile  Library, 


San  Francisco,  Mar.  1,  1885. 
To  James  Land,  Dr. 


To  binding 

27  vol. 

Atlantic  Monthly, 

.     @  900 

2     '' 

Pop.  Sci.  Mo., 

.     ''   900 

3     " 

St.  Nicholas,    .     . 

.     "   750 

1     " 

Overland  Mo., 

3     '' 

Harper's  Mo., 

.     ''   900 

4     '' 

Century,      .     .     . 

.     "   750 

90 


3. 

Mr.  H.  B.  Crockett, 


HoLLiSTER,  Cal.,  Sept.  3,  1884. 
Bought  of  Smith  &  Tyler. 


1884 

June 

o 

a 

i  i 

July 

5 

(I 

n 

ii 

17 

Aug. 

10 

<  ( 

13 

11 

28 

5  Gent's  Collars, @       300 

1  doz.  Hdkf.,       ''        250 

2  pr.  Kid  Gloves, ^'  $1.75 

3  doz.  Buttons, "        400 

3  Fine  Linen  Shirts,     .     .     .     .  "  $2.25 

4  pr.  Gent's  Hose, "       350 

3  pr.  Linen  Cuffs, "        400 

1  Derby  Hat, 


50 


ARITHMETIC.  121 

EXERCISE    172. 

Make  out  bills  of  the  following,  and  receipt  them,  using 
your  own  and  classmates'  names: 

1.  5  yd.  Ribbon  @  12^  cents.  11  yd.  Black  Cashmere  @ 
$1.60.  4  doz.  Buttons  @  30  cents.  2  yd.  Silicia  @  20  cents. 
10  yd.  Sheeting  @  18  cents.     1  pr.  Gaiters  $3.50. 

2.  5  gal.  Kerosene  Oil  @  25  cents.  3  pr.  Blankets  @  $6.50. 
25  lb.  Brown  Sugar  @  7  cents.  3  doz.  Eggs  @  20  cents.  1 
Turkey,  12  lb.,  @  22  cents.    50  lb.  Irish  Potatoes  @  1^  cents. 

3.  Mar.  3,  1880,  2  lb.  Steak  @  12^  cents.  Mar.  4,  4^  lb. 
Roast  Beef  @  12  cents.  March  5,  If  lb.  Sirloin  @  15  cents. 
March  6,  5^  ft.  Mutton  @  10  cents.  Mar.  8,  1  15-ft.  A..&  C. 
Ham  @  19  cents.     Mar.  10,  3  ft.  Veal  Roast  @  14  cents. 

4.  John  Smith  performed  12  days'  work  for  M.  S.  John- 
son at  $1.50  per  day. 

5.  \  doz.  Wooden  Chairs  @  $1.  1  Lounge  $12.50.  1  Bed 
Room  Set  $22.75.  3  Fancy  Chairs  @  $2.25.  1  Extension 
Table  $7.50.     1  Center  Table  $4. 

6.  S.  Wilson  sold  Geo.  Sims  10  tons  of  hay  @  $10  a  ton. 

7.  8  doz.  Oranges  @  15  cents.  10  ft.  Nuts  @  10  cents. 
8  Lemons  @  2|  cents.  5  ft.  Mixed  Candies  @  20  cents. 
1  box  Apples  $1.     7  boxes  Strawberries  @  45  cents.  ' 

8.  1  doz.  Lead  Pencils  @  5  cents  each.  \  ream  Note 
Paper  40  cents.  4  Note  Books  @  10  cents.  1  Rubber 
Eraser  5  cents.  1  package  Envelopes  10  cents.  2  Fifth 
Readers  @  85  cents.     2  School  Geographies  @  $1.40. 

9.  14  yd.  Print  @  12  cents.  3  ft.  Butter  @  28  cents.  4 
bars  Soap  @  10  cents.  1  pr.  Child's  Shoes  $1.75.  25  ft. 
Flour  @  2\  cents.  1  can  Lard  65  cents.  2  ft.  Cheese  @ 
17  cents. 

EXERCISE  173. 

Ask  your  parents  for  3  bills  they  may  have,  and  bring  to 
the  class  for  dictation.  They  may  caution  you  not  to  lose 
them  even  though  paid.     Why? 


122  CALIFORNIA   SERIES. 


WEIGHTS  AND  MEASURES. 

A  concrete  number  when  written  in  terms  of  one  denom- 
ination is  simple ;  thus, 

125  yards ;  15.25  dollars;  3.435  hours  are  simple  numbers. 

But  when  expressed  in  two  or  more  different  units  it  is 
said  to  be  compound ;  thus, 

2  yards  2  feet  3  inches  is  a  compound  number. 

These  compound  numbers  were  used  before  the  decimal 
notation  was  known  to  English-speaking  people.  Calcula- 
tions are  made  much  simpler  and  easier  by  the  use  of 
decimals. 

The  compound  number  has  properly  no  unit ;  although 
one  of  any  denomination  may  be  taken  as  the  unit.  In 
the  example  given  above  of  2  yd.  2  ft.  3  in.,  the  unit  may 
be  1  yd.,  1  ft.,  or  1  in. 

15.6  pounds:  7  lb.  6  oz.;  16  ounces;  17  gal.  3  qt.  1  pt.; 
13.75  inches;  156  A.  25  rd.;  13  quarts;  7  bu.  3  pk.  7  qt.  1  pt. 

From  the  above  select  the  simple  and  the  compound 
numbers,  and  analyze  in  each  case,  thus: 

1.  is  a  simple  number  because  it  is  a tvrit- 

ten  in of  one . 

2.  is  a  compound  number  because  it  is writ- 
ten in or . 


LOI^G  OE  LINEAK  MEASUEE. 

The  denominations  of  length  or  line  measurement  are  given 
in  the  following  table: 

12  inches  (m.)  =  l  foot  (ft.). 
3  ft.  =1  yard  (yd.). 

b}4yd.  =lrod(rd.). 

320  rd.  :-l  mile  (mi.). 


ARITHMETIC. 


123 


In  the  4-inch  scale  here  given  what  divisions  of 
the  inch  do  you  find? 

Draw  on  paper  a  hne  2|-  in.  long. 

Draw  a  Une  that  you  think  is  3|  in.  long. 

Measure  it. 

Draw  on  the  blackboard  a  line  1  ft.  long;  1|  ft. 
long;  1  yd.  long.      . 

Divide,  without  measuring,  the  last  line  into 
parts  each  9  in.  long. 

Now  measure  them. 

Divide  your  yard  line  into  3  equal  parts.  How 
long  should  each  part  be?     Measure. 

Draw  the  same  line  vertical  or  oblique  and 
divide  it  into  6  in.  parts. 

Estimate  the  width  and  length  of  your  desk; 
measure  it.  The  width  and  height  of  the  win- 
dow; the  door;  dimensions  of  the  blackboard; 
dimensions  of  the  teacher's  desk;  measure  each. 

Estimate  the  length  and  width  of  your  scbool- 
room,  and  measure,  using  the  yardstick  or  tape 
measure.' 

Pace  off  the  length  and  width  of  your  school 
yard,  and  then  measure  with  the  tape-line.  Take 
long  steps.  Eind  from  this  how  long  your  paces 
are. 


A  l^right,  intelligent  class  well  fitted  in  the  work  thus  far,  will 
need  the  pencil  or  crayon  to  record  results  only  in  most  of  the  fol- 
lowing work. 

EXERCISE    174. 

1.  How  many  in.  in  1^  yd.?     In  2  yd.? 

2.  How  many  ft.  in  7  yd.?     In  4  yd.? 

3.  How  many  yd.  in  3  rd.  ?     In  4  rd.? 

4.  In  108  in.  how  many  ft.?     In  720  in.  ? 

5.  What  part  of  a  mi.  are  16  rd.?     160  rd.  ? 

6.  What  part  of  a  ft.  are  9  in.?     6  in.?     4  in.? 


124  CALIFORNIA   SERIES. 

7.  What  part  of  a  yard  are  2  ft.  ?     9  in.  ?     18  in.  ? 

8.  In  "2  mi.  how  many  rd.? 

9.  In  -J  yd.  how  many  in.? 

10.  In  -g  mi.  how  many  yd.  ? 

11.  2  ft.  6  in.  are  what  part  of  a  yd.? 

12.  How  many  rd.  in  1  mi.?     1  mi.  25  rd.? 

13.  How  many  yd.  in  5  rd.?     In  5  rd.  3  yd.? 

14.  How  many  yd.  in  1  mi.?     How  many  ft.  in  1  mi.? 
How  many  in.? 

15.  In  1  mi.  20  rd.  4  yd.  how  many  yd.? 

16.  How  many  inches  in  1  rd.  2  yd.  2  ft.  7  in.? 

17.  How  many  ft.  are  there  in  2  mi.  35  rd.  2  yd.  2  ft.? 

18.  How  many  ft.  and  in.  are  there  in  40  in.? 

19.  How  many  yd.  ft.  and  in.  are  there  in  79  in.? 

20.  How  many  rd.  yd.  ft.  and  in.  are  there  in  607  in.? 

21.  How  many  yd.  ft.  and  in.  are  there  in  874  in.? 

22.  How  many  yd.  ft.  and  in.  in  211  in.?     In  100  in.? 

23.  How  many  rd.  yd.  and  ft.  in  373  ft.? 

24.  HoAV  many  rd.  are  there  in  35  yd.? 

25.  How  many  mi.  and  rd.  are  there  in  650  rd.? 

26.  How  many  rd.  and  yd.  are  there  in  98  yd.? 

27.  How  many  rd.  yd.  and  ft.  in  1000  ft.?     In  179  ft.? 

To  reduce  to  a  decimal  of  any  denomination. 

28.  Reduce  3  mi.  235  rd.  2  yd.  2  ft.  3  in.  to  rd. 


12 

3  in. 

3 

3 

2.25  ft. 

adding  the  2  ft. 

5^ 

2.7  5  yd. 

adding  the  2  yds. 

2  3  5.5  rd. 

960.0     =: 

adding  the  235  rds 
mi. 

1  19  5.5rd. 

29.  Express  2  ft.  6  in.  in  inches.     Express  decimally, 
using  the  ft.  as  the  unit. 


ARITHMETIC.  125 

30.  Express  3  mi.  2  rd.  4  yd.  2  ft.  in  ft.     Express  deci- 
mally, using  the  yd.  as  the  unit. 

31.  Express  3  yd.  2  in.  as  in.     Express  decimally,  with 
the  yd.  as  the  unit. 

32.  Express  1  mi.  2  rd.  2  ft,  in  ft.     Express  decimally, 
with  the  mi.  as  the  unit. 

33.  Express  3  rd.  4  yd.  2  ft.  6  in.  in  inches.     Express 
decimally,  using  the  ft.  as  the  unit. 

34.  Express  2  rd.  1  yd.  2  ft.  6  in.  as  in.     Express  deci- 
mally, using  the  ft.  as  the  unit. 

35.  Express  1  mi.  2  rd.  1  yd.  1  ft.  6  in.  as  in.     Express 
decimally,  using  the  yd.  as  the  unit. 

36.  Express  3  mi.  80  rd.  in  ft.     Express  decimally,  with 
the  mi.  as  the  unit. 

37.  Express  2  mi.  2  rd.  3  ft.  in  ft.     Express  decimally, 
with  the  mi.  as  the  unit. 

38.  Express  3  rd.  2  yd.  2  ft.  3  in.  in  in.     Express  deci- 
mally, with  the  rd.  as  the  unit. 

39.  Express  4  mi.  240  rd.  as  yd.     Express  decimally, 
with  the  mi.  as  the  unit. 

40.  Express  3  mi.  8  rd.  3  yd.  2  ft.  3  in.  in  ft.     Express 
decimally,  with  the  yd.  as  the  unit. 

41.  Express  7  rd.  2  yd.  2  ft.  3  in.  as  in. 

To  reduce  to  a  fraction  of  higher  denomination. 

42.  3  yd.  2  ft.  6  in.  is  what  fraction  of  a  rd.? 

3  yd.  2  ft.  6  in. 


6  in.=^  ft. 
3" 


2i  ft.=:^=f  yd. 


^  ycl.=i=ll  rd 


43.  Change  4|-  rd.  to  a  fraction  of  a  mi. 

44.  Reduce  -f  mi.  to  a  decimal  of  a  mi. 


126  CALIFORNIA   SERIES.  t 

45.  Reduce  .375  mi.  to  rd. 

46.  2  ft.  6f  in.  are  what  decimal  of  a  rd.? 

47.  Change  65  rd.  2  yd.  2  ft.  6  in.  to  the  decimal  of  a  mi. 

48.  Express  25  rd.  4^  yd.  as  a  decimal  of  35  rd.  3  yd. 
2|ft. 

49.  Express  42  rd.  2  yd.  4.3  in.  as  a  decimal  of  a  mi. 

50.  Express  6  ft.  8.5  in.  as  a  decimal  of  a  rd. 

51.  Express  S^  yd.  as  a  fraction  of  7  yd.  4  in. 

52.  Express  165  rd.  2  yd.  2  ft.  9  in.  as  a  fraction  of  a  mi. 

53.  Express  2  yd.  2  ft.  2  in.  as  a  fraction  of  3  yd. 

54.  Express  98  rd.  7  yd.  2  ft.  4  in.  as  a  fraction  of  a  mi. 

EXERCISE   175. 
Bring  in  10  examples  of  your  own  like  those  of  the  pre- 
ceding exercise,  for  dictation. 


SUEYEYOE^S  LOT^G  MEASUEE. 

The  land  surveyor  uses  the  following  table.  It  has  some 
advantages  over  the  table  given  above,  being  partly  deci- 
mal. 

25  links  (l.)  =  l  rd. 

4  rd.  —1  chain  (ch.). 

80  ch.  =1  mile  (mi.). 

EXERCISE  176. 

1.  How  many  ch.  in  2  mi.?     In  3|  mi.? 

2.  How  many  links  in  4  rd. ?     In  5  rd.? 

3.  How  many  rd.  in  7  ch. ?     In  5A  ch.? 

4.  How  many  yd.  in  2  ch.  ? 

5.  How  many  ch.  in  |  of  a  mi.? 

6.  In  160  ch.  how  many  mi.?  In  320  ch.?  In  96  ch.? 
In  100  ch.? 

7.  What  part  of  a  ch.  are  75  1.  ?     20  1.? 

8.  What  part  of  a  mi.  are  5  rd.?     16  rd.? 


ARITHMETIC. 


\Z 


9.  What  part  of  a  rd.  are  5  1.  ?     10  1.  ? 
10;  Reduce  1  mi.  2  ch.  1  rd.  to  1. 

11.  Reduce  2  mi.  2  rd.  6  1.  to  1. 

12.  Reduce  5  mi.  79  ch.  8  rd.  to  rd. 

13.  Change  29763  1.  to  higher  denominations. 

14.  Change  8543  1.  to  higher  denominations. 

15.  How  many  mi.  ch.  rd.  and  1.  in  79G328  1.  ? 

16.  How  many  mi.  ch.  rd.  and  1.  in  76543  1.? 

17.  Reduce  5  ch.  15  L  to  rd. 

18.  What  part  of  a  rd.  is  15  1.? 

19.  How  many  rd.  in  3^  ch.?     How  many  1.? 

20.  Reducelmi.  Ich.  Ird.  11.  toch.   To  rd.  To  mi.  Tol. 

21.  What  is  the  difference   between   15  ch.   44  1.   and 
15.44  ch.?  • 


LONG  MEASUEE-METEIO  SYSTEM. 

The  French  express  measures  and  weights  decimally, 
and  their  methods  and  measuring  units  have  been  adopted 
by  most  of  the  nations  of  Europe.  In  the  United  States  this 
system  has  been  legalized,  and  is  coming  slowly  into  use. 

The  unit  of  length  is  one  ten-millionth  the  meridian  dis- 
tance from  the  equator  to  the  pole;  is  nearly  39.37  inches, 
and  is  called  a  meter,  whence  this  system  is  called  the 
metric  system. 

The  decimal  places  have  received  names;  thus. 


tz 

a 

C^ 

iJ, 

^h' 

?-l 

-M 

'r-l 

:v 

%^ 

o 

<v 

o 

(V 

'CD 

-^ 

Qj 

-^^ 

-M 

>H 

-M 

o 

•^ 

<D 

CD 

.r— I 

0) 

O 

•t— 1 

•  I— 1 

g 

2 

•  1— 1 

a 

s 

0000     0.0     0     0 


Qi  (That  myria,  kilo,  hekto,  cleka,  deci,  centi,  milli,  are 

(     prefixes  used  in  weights  and  measures. 


128 


CALIFORNIA   SERIES. 


EXERCISE   177. 

1.  How  many  centimeters  are  there  in  47.265 
meters?  How  many  decimeters?  How  many 
dekameters  ? 

2.  Read  3825.386  meters,  by  placing  in  turn 
each  of  its  names  in  place  of  meters  and  with- 
out changing  its  value. 

3.  In  the  above  metric  scale  what  is  the  dis- 
tance from  "  c  "  to  "  s  "? 

4.  AVrite  the  distance  from  "  d  "  to  each  of 
the  following  points:  a,  n,  q,  r,  h,  t. 

5.  AVrite   the   distance   from  "  x "   to  each 
^    of    the    following   points:    v,  t,  o,  1,  h,  m,  b, 

a,  q,  g. 

6.  Draw  on  paper  a  line  .187  meters  long. 

7.  Draw  on  the  blackboard  a  line  1  meter 
long. 

8.  Divide  this  line  into  halves;  into  tenths. 
What  are  the  last  divisions  called  ? 

9.  How  many  feet  and  inches  in  5.24 
meters  ? 

10.  How    many   yd.    ft.    and   in.    in    35.428 
meters  ? 

11.  In  5785  meters  how  many  mi.  ? 

12.  In  7856918  in.  how  many  meters? 


■^^ 

z 

2 

= 

X 

— 

— 

— 

V 

I 
s 
r 

1 
P 

0 

n 
m 

— 

— 

— 

a 

— 

— 

— 

I 

't-i 

o 

— 

— 

h 

— 

— 

— 

h 
9 
f 
e. 

— 

— 

— 

d 
c 
b 
a 

SUEFACE  MEASUEE. 

A  flat  surface  of  four  straight  edges  and  square  corners 
is  a  rectangle. 

A  square  is  a  rectangle  with  equal  edges. 

The  measuring  unit  is  a  square  having  a  linear  unit  for 
its  edge  :  as  1  square  foot,  1  square  meter,  1  square  inch,  1 
square  mile. 


ARITHMETIC. 


129 


TABLE. 

144      square  (sq.)  in.  =  l  sq.  ft. 
9      sq.  ft.  =  1  sq.  yd. 

30)^  sq.  yd.  =1  sq.  rd. 

160      sq.  rd.  =1  acre.  (A.) 

640      A.  =1  sq.  mi.  or  section  of  land. 

The  side  or  edge  of  the  acre  is  not  a  unit  of  lensrth. 


A  rectangle  6  units 
long  and  1  unit  wide 
contains  6  square  units; 
3  units  wide  it  contains 
3  times  6  sq.  units,  or 
18  sq.  units;  hence, 


2 


To  find  the  area  of  a  rectangle. 

f  1st. — The  length  and  the  breadth  are  factors  of 
Observe.  \  the  area. 

[^  2d. — The  midtiplier  is  an  abstract  number. 


EXERCISE    178. 

1.  A  rectangle  8  in.  long  and  5  in.  wide  contains  how 
many  sq.  in.  ? 

2.  How  many  sq.  ft.  in  a  rectangle  17  ft.  long  and  13 
ft.  wide? 

3.  How  many  sq.  yd.  in  a  square  whose  edge  is  36  ft.? 

4.  How  many  sq.  in.  in  2  sq.  ft.  ? 

5.  In  6  sq.  yd.  how  many  sq.  ft.  ? 

6.  In  3  A.  how  many  sq.  rd.  ? 

7.  In  1^  sq.  mi.  how  man}^  sq.  rd.  ? 

8.  In  36  sq.  ft.  how  many  sq.  yd.  ? 

9.  How  many  sq.  ft.  in  288  sq.  in.  ? 

10.  In  2^  acres  how  many  sq.  rd.? 

11.  How  many  sq.  ft.  in  a  table  5^  ft.  long  and  3  ft.  wide? 

9— A 


130  CALIFORNIA   SERIES. 

12.  If  your  reader  is  7^  in.  long  and  5  in.  wide,  how  many 
sq.  in.  in  the  surface  of  its  sides  ? 

13.  How  many  sq.  in.  in  |  of  a  sq.  ft.? 

14.  How  many  sq.  ft.  in  f  of  a  sq.  yd.? 

15.  What  cost  a  quarter  section  of  land  at  $1.25  an  acre? 

16.  How  many  sq.  yd.  in  2  sq.  rd.? 

17.  In  a  piece  of  land  9  ft.  wide  and  12  ft.  long,  how 
many  sq.  yd.? 

18.  Reduce  1  sq.  rd.  to  sq.  in.     One  A.  to  sq.  ft. 

19.  Change  2  A.  40  sq.  rd.  17  sq.  ft.  to  sq.  ft. 

20.  How  many  sq.  ft.  in  3  A.? 

21.  Find  the  number  of  sq.  yd.  in  3  sq.  mi.,  17  sq.  rd.,  and 
4  sq.  yd. 

22.  Find  the  number  of  sq.  ft.  in  3476  sq.  in. 

23.  How  many  sq.  ft.  and  in.  in  98756  sq.  in.? 

24.  Change  7856  sq.  ft.  to  higher  denominations. 

25.  Reduce  48413  sq.  yd.  to  higher  denominations. 

26.  At  $75|  an  A.,  what  is  the  value  of  a  farm  189.5  rd. 
long  and  150  rd.  wide? 

27.  If  37  A.  128  sq.  rd.  are  uncultivated  in  a  farm  of  170 
A.  16  sq.  rd.,  what  part  of  the  farm  is  cultivated? 


SURVEYOR'S  SURFACE  MEASURE. 

625  sq.  1.   =  1  sq.  rd. 

16  ''  rd.  =  l  "    ch. 

10  "  ch.  =  l  A. 
640  A.        =1  sq.  mi.  or  section. 

EXERCISE   179. 

1.  How  many  sq.  rd.  in  3  sq.  ch.?     In  2\  sq.  ch.  how 
many  sq.  rd.? 

2.  How  many  sq.  ch.  in  5  A.  ?     In  6^  A.? 

3.  In  1  sq.  mi.  how  many  sq.  ch.  ? 

4.  How  many  sq.  ch.  in  \  of  an  A.  ? 


ARITHMETIC.  131 

5.  Twenty-five  sq.  1.  are  what  part  of  a  sq.  rd.? 

6.  Eight  sq.  rd.  are  what  part  of  a  sq.  ch.? 

7.  What  part  of  an  A.  are  5  sq.  ch. ?     2  sq.  ch.? 

8.  80  A.  are  what  part  of  a  sq.  mi.  ? 

9.  In  a  quarter  section  land,  how  many  A.? 

10.  In  a  section  of  land,  how  many  sq.  1.? 

11.  Reduce  160  A.  to  sq.  1. 

12.  How  many  sq.  ch.  in  2  sq.  mi.  6  A.  9  sq.  ch.? 

13.  Reduce  an  A.  to  sq.  1. 

14.  In  1  sq.  mi.  1  A.  1  sq.  ch.  1  sq.  rd.,  how  many  sq.  1.? 

15.  Change  842590  sq.  1.  to  higher  denominations. 

16.  Reduce  25373896  sq.  1.  to  higher  denominations. 

17.  In  98754  sq.  rd.,  how  many  A.,  sq.  ch.,  and  rd.  ? 

18.  In  9857  sq.  ch.,  how  many  sq.  mi..  A.,  and  ch.? 

19.  Reduce  75328  sq.  rd.  to  higher  denominations. 

20.  A  man  owned  a  piece  of  land  46  ch.  long  by  37  ch. 
wide;  he  sold  a  piece  containing  42  A.  5^  sq.  ch.;  what 
part  of  the  whole  was  left  ? 

21.  A  man  owns  a  piece  of  land  containing  12  A.;  has 
an  irrigating  ditch  cut  through  one  part  of  it,  25  1.  wide  and 
5  ch.  long;  what  part  of  an  A.  does  he  lose  by  the  ditch, 
and  what  part  of  the  whole  can  he  cultivate? 

22.  A  surveyor,  starting  at  a  certain  point,  laid  his  chain 
10.6  times  to  the  east,  then  south  5  times,  then  west  5.3 
times,  then  south  3  times,  then  west  5.3  times,  then  meas- 
ured north  to  the  place  of  beginning;  how  long  was  his  last 
line,  and  how  many  acres  in- the  field? 


SURFACE  MEASURE— METRIC  SYSTEM. 

The  unit  of  land  measure  is  a  square  whose  edge  is  10 
meters;  hence  the  unit  contains  100  square  meters.  The 
name,  are,  meaning  area,  has  been  given  to  it. 

All  perfect  squares,  in  the  decimal  system,  end  in  the 


132 


CALIFORNIA   SERIES. 


alternate  places  commencing  with  units;  hence  only  those 
places  have  received  names. 


Small  areas  are  given 
in  square  meters  or  centi- 
ares,  while  large  ones  are 
usually  given  in  hektares. 

The  hektare:=2.47  acres, 
nearly. 


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EXERCISE    180. 

1.  236.47925  ares:  read  this  number  as  hektares;  ascen- 
tiares;  as  square  meters;  as  square  decimeters;  as  square 
centimeters. 

2.  In  34652  sq.  centimeters,  are  how  many  ares? 

3.  How  much  land  in  a  field  234.56  meters  long  and 
184.25  meters  wide  ? 

4.  What  is  the  difference  between  6  square  meters  and 
6  meters  square  ? 

5.  How  many  ares  in  1  quarter  section  of  land? 


CARPETING. 


The  lighter  carpets,  as  ingrains,  are  1  yard  wide;  the 
heavier,  as  Brussels,  etc.,  are  |  yd.  wide.  Carpets  waste  in 
matching,  according  to  the  pattern,  the  breadths  requiring 
to  be  cut  longer  than  tlie  room. 

EXERCISE    181.    (Written.) 

1.  How  many  yd.  of  carpeting  |  yd.  wide  will  it  take  for 
a  room  17  ft.  long,  15  ft.  9  in.  wide,  if  the  breadths  run 
lengthwise? 

2.  How  nuich  yard-wide  carpeting  will  be  required  for  a 
room  17  ft.  6  in.  wide,  23  ft.  4  in.  long,  if  the  breadths  run 
crosswise  ?     If  they  run  lengthwise  ? 


ARITHMETIC.  183 

^^^^ 

3.  How  many  yd-  of  carpeting  |  yd.  wide  will  it  take  for 
a  room  11  ft.  wide,  15  ft.  long,  if  the  breadths  run  across 
the  room?     If  they  run  lengthwise? 

^^C^  What  will  it  cost  to  carpet  a  room  19  ft.  wide  and  24 
it.  long,  with  carpet  a  yd.  wide  costing  $1.25  per  yd.,  the 
breadths  to  run  crosswise  and  ^  of  a  yd.  on  each  breadth 
allowed  for  matching? 

5.  Find  the  smallest  cost  at  which  a  room  can  be  car- 
peted, which  is  13  ft.  G  in.  wide,  18  ft.  long,  the  carpet  |  yd. 
wide  at  $2.75  per  yd.,  \  yd.  in  each  breadth  allowed  for 
waste  in  matching. 

PLASTERING. 

Usually  estimated  by  the  square  yard.  Multiply  the 
distance  around  the  room  by  the  height  from  floor  to  ceil- 
ing ;  deduct  \  the  surface  of  the  openings  and  add  the  area 
of  the  ceiling. 

6.  At  $.27  per  sq.  yd.  what  will  it  cost  to  plaster  a  room 
18  ft.  wide,  20  ft.  long,  10  ft.  high,  having  3  windows,  each 

2  ft^  wide,  6  ft.  high,  and  1  door  3  ft.  wide,  7  ft.  high? 
^jfC^How  many  sq.  yd.  of  plastering  in  17  rooms  of  a  hotel, 

each  11  ft.  wide,  12  ft.  long,  12  ft.  high,  there  being  T  win- 
dow 2\  ft.  wide  and  6  ft.  high,  and  1  door  2f  ft.  wide  and 
7  ft.  high  in  each  room  ? 

8.  How  many  sq.  yd.  of  plastering  in  a  hall  90  ft.  long, 
65  ft.  wide,  and  24  ft.  high,  there  being  13  windows,  each 

3  ft.  wide  and  10  ft.  high,  and  4  doors,  each  4  ft.  wide  and 
9  ft.  high? 

9.  How  many  yd.  of  plastering  in  a  room  16  ft.  wide,  24 
ft.  long,  and  9  ft.  high,  allowing  12  sq.  yd.  for  doors  and 
windows  ? 

10.  What  will  it  cost  to  plaster  a  room  24  ft.  6  in.  long, 
15  ft.  3  in.  wide,  and  10  ft.  high,  at  $.30  per  sq.  yd.,  allow- 
ing 14  sq.  yd.  for  doors  and  windows? 

11.  Find  the  cost  of  papering  a  room  15  ft.  wide,  18  ft. 


134  CALIFORNIA   SEHIES. 


long,  and  10  ft.  high,  with  paper  24  in.  wide  at  $.95  a  roll, 
8  yd.  in  a  roll,  20  sq.  yd.  allowed  for  doors  and  windows. 
.1^^.  Find  the  cost  of  plastering  a  room  17  ft.  6  in.  wide, 
24  ft.  8  in.  long,  10  ft.  high,  at  $.33  per  sq.  yd.,  allowing  50 
sq.  yd.  for  doors  and  windows. 

*j4s^^hat  will  be  the  cost  of  the  paper  for  the  same  room, 
18  in.  wide,  8  yd.  in  a  roll,  at  $.75  a  roll? 

14.  How  many  thousand  shingles  will  it  take  to  cover  a 
roof  whose  rafters  are  25  ft.  long,  and  ridge  pole  30  ft.  long, 
if  4  in.  in  width  and  5  in.  in  length  of  each  shingle  is 
exposed  to  the  weather? 

15.  How  many  bricks  8  in.  long,  4  in.  wide,  will  be 
required  for  a  sidewalk  100  ft.  4  in.  long  and  4  ft.  wide? 

**¥&.  If  it  takes  840  sheets  of  tin  16  in.  wide  and  24  in. 
long,  to  roof  a  house,  what  is  the  area  covered? 

17.  AVhat  will  it  cost  to  plaster  a  room  18  ft.  long,  16  ft. 
wide,  and  12  ft.  high,  at  $.37^  per  sq.  yd.,  allowing  for  3 
windows  2|  by  8  ft.  and  2  doors  3  by  8  ft. 

18.  In  a  border  of  tiling  around  my  fireplace,  8  in.  w'ide, 
3  ft.  8  in.  high,  and  4  ft.  across  the  top,  how  many  tiles  4 
in.  sq.?     Draw  diagram  before  working. 

EXERCISE   182. 

Measure  the  length,  breadth,  and  height  of  all  the  rooms 
in  your  school  building;  also  take  the  dimensions  of  the 
windows,  doors,  and  baseboards.  Estimate  the  cost  of 
carpeting  and  plastering  each  room. 

Make  the  same  measurements  of  three  rooms  at  your 
homes  and  bring  to  the  class  for  similar  work. 


SOLID  OE  CUBIC  MEASUEE. 

A  solid,  all  whose  faces  are  rectangles  or  squares,  is  a 

rectangular  solid ;  the  cube  has  equal  square  faces.  I 

The  unit  of  solid  or  cubic  measure  is  a  cube  having  a 


ARITHMETIC. 


135 


linear  unit  for  its  edge;  as  1  cubic  inch,  1  cubic  yard,  1 

cubic  meter. 

Work. — This  rectangular  solid  is 
4  units  long,  3  units  wide,  and  3  units 
high.  How  many  cubic  units  does 
it  contain? 

4  longXS  wide  gives  12  sq.  units  in 
base ;  for  1  unit  high  there  are  12  cubic 
units,  and  for  3  units  high  3X12  cubic 
units=36  cubic  units. 

1. — The  length,  width,  o»r/  height  are  factors  of 
Observe.  "!  the  cubic  contents. 

2. — The  multipliers  are  abstract  numbers. 

Do  the  same  w^ork  with  the  foot  as  the  unit;  with  the 
yard;  with  the  meter;  with  the  inch. 

TABLE. 

1728  cu.  in.=l  cu.  ft. 
27  cu.  ft.  =1  cu.  yd. 
128  cu.  ft.  =1  cd. 


EXERCISE   183. 

1.  How  many  cu.  ft.  in  a  room  17^  ft.  long,  14  ft.  wide, 
and  12  ft.  high?     How  many  cu.  yd.? 

2.  How  many  cd.  of  wood  are  in  a  pile  18  ft.  long,  12 
ft.  wide,  and  9^  ft.  high? 

3.  How  many  cu.  ft.  in  ^  cu.  yd.? 

4.  How  many  cu.  in.  in  a  brick  a  ft.  long,  5  in.  wide, 
and  1^  in.  thick? 

5.  How  many  cu.  ft.  in  2  cu.  yd.? 

6.  How  many  cu.  ft.  in  1^  cd.  of  wood? 

7.  How  many  cu.  ft.  in  3  cu.  yd.? 

8.  How  many  cu.  yd.  in  108  cu.  ft.? 

9.  What  part  of  a  cu.  yd.  are  9  cu.  ft.  ? 

10.  What  part  of  a  cd.  are  64  cu.  ft.? 

11.  Four  cu.  ft.  are  what  part  of  a  cd.  ? 


136  CALIFORNIA   SERIES. 

12.  How  many  cu.  ft.  in  a  stick  of  timber  9  in.  wide,  4 
in.  thick,  and  24  ft.  long? 

13.  A  trench  for  a  water  main  is  3  ft.  deep,  2  ft.  wide, 
and  21  ft.  long;  how  many  cu.  ft.  have  been  excavated  to 
form  it? 

14.  How  many  cd.  in  a  pile  of  wood  4  ft.  wide,  4  ft.  high, 
and  8  ft.  long? 

15.  How  many  cu.  ft.  in  a  tank  8  ft.  long,  6  ft.  wide,  and 
3  ft.  deep? 

16.  Change  13  cu.  yd.  11  cu.  ft.  to  cu.  in. 

17.  In  9  cu.  yd.  4  cu.  ft.  13  cu.  in.  how  many  cu.  in.? 

18.  Change  159728  cu.  in.  to  higher  denominations. 

19.  If  there  are  9  cu.  ft.  828  cu.  in.  in  one  block  of  stone, 
and  7  cu.  ft.  932  cu.  in.  in  another,  how  many  in  both? 

20.  How  many  cu.  yd.  in  a  room  12  ft.  long,  11  ft.  wide, 
and  9  ft.  high? 

21.  A  parcel  of  wrapping  paper  is  30  in.  long,  24  in.  wide, 
and  2  in.  thick;  how  many  cu.  in.  does  it  contain? 

22.  If  a  cu.  ft.  of  stone  weighs  175  ft).,  what  will  be  the 
weight  of  a  cu.  yd.? 

23.  A  water  tank  is  7  ft.  deep,  9  ft.  long,  and  7  ft.  wide. 
Find  its  contents  in  cu.  in. 

24.  If  a  load  of  wood  is  3|  ft.  wide  and  5  ft.  high,  how 
long  must  it  be  to  contain  1-J  cd.? 

25.  A  block  of  stone  is  7  ft.  in  each  dimension;  how 
many  cu.  yd.  does  it  contain? 

26.  How  many  cd.  of  wood  in  a  pile  56  ft.  long,  4^  ft. 
high,  and  6  ft.  wide? 

27.  A  dealer  gets  $6.50  a  cd.  for  a  pile  of  wood  16  ft. 
long,  4^  ft.  wide,  and  7  ft.  6  in.  high;  how  much  does  he 
receive  ? 

28.  In  247  cu.  ft.  how  many  cu.  yd.? 

29.  The  excavation  for  a  block  of  stores  was  63  ft.  wide, 
157  ft.  long,  and  8  ft.  deep;  how  many  cu.  yd.  of  earth 
were  excavated? 


ARITHMETIC. 


ISI 


30.  Find  the  value  of  the  wood  piled  on  one  half  of  a 
vacant  lot  60  ft.  by  150  ft.,  the  wood  being  piled  9  ft.  high, 
and  Avorth  $9.50  per  cord. 

31.  Reduce  20  cu.  ft.  432  cu.  in.  to  the  decimal  of  a  cu.  yd. 

32.  216  cu.  in.  are  what  decimal  of  a  cu.  ft.? 

33.  648  cu.  in.  are  what  fraction  of  a  cu.  yd.? 

34.  Reduce  .75  of  a  cu.  yd.  to  cu.  in. 

35.  What  is  .975  of  a  cu.  yd.  expressed  in  lower  denomi- 
nations? 

36.  Express  .375  of  a  cd.  in  cu.  ft. 

37.  How  many  cu.  ft.  of  air  does  your  school  room  contain? 


SOLID  MEASUEE-METEIC  SYSTEM. 

In  the  decimal  notation  of  solid  measure  the  stere,  or 
cubic  meter,  is  the  unit. 


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EXERCISE   184. 

The 


stere=1.308  cubic 
yards,  or  .276  cords. 

1.  Read  34.6  steres  as 
decisteres;  as  dekasteres. 

2.  Read  225.463829731 
cubic  meters  as  cubic 
decimeters;  as  dekasteres; 
as  cubic  millimeters;  as 
cubic  centimeters. 

3.  In  28.5  steres  of  wood  are  how  many  cd.  ? 

4.  How  much  wood  in  a  pile  7.2  meters  long,  1.7  meters 
wide,  and  2  meters  high  ? 


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STONE  AND  BRICK  WORK. 

A  wall  is  measured  on  the  outside,  no  allowance  being 
made  for  corners.     All  measurements  are  made  in  feet. 


138  CALIFORNIA   SERIES. 

Multiply  the  length  thus  obtained  by  the  height,  deduct 
the  surface  of  the  openings,  and  multiply  by  the  thickness; 
divide  by  16^  for  perches,  or  multiply  by  21  for  the  number 
of  bricks. 

EXERCISE    185. 

1.  At  $5.25  a  perch  what  will  it  cost  to  build  a  wall 
around  a  piece  of  land  22  ft.  by  45  ft.,  the  wall  1^  ft.  thick 
and  8  ft.  high? 

2.  How  many  bricks  would  it  take  to  build  the  walls  of 
a  house  30  ft.  wide,  45  ft.  long,  and  20  ft.  high,  the  wall  to 
be  1  ft.  thick?  There  are  10  windows,  each  2^  ft.  wide  and 
7  ft.  high,  and  4  doors,  each  3  ft.  wide  and  8  ft.  high. 

3.  At  $4  per  thousand  for  bricks,  what  will  it  cost  for  the 
wall  of  a  building  58  ft.  long,  25  ft.  wide,  44  ft.  high,  the 
wall  to  be  U  ft.  thick;  there  being  20  windows,  each  3  ft. 
wide,  8  ft.  high,  and  9  doors,  each  3  ft.  wide  and  8  ft.  high? 

4.  How  many  perches  of  masonry  in  a  wall  5  ft.  high,  1^ 
ft.  thick,  inclosing  a  garden  9  rd.  long,  7  rd.  wide? 

5.  How  many  perches  of  stone  in  a  wall  2  ft.  thick  and 
4  ft.  high,  inclosing  a  piece  of  land  40  rd.  sq.  ? 

6.  How  many  bricks  will  it  take  to  build  a  house  46  ft. 
long,  34  ft.  wide,  20  ft.  high,  the  wall  18  in.  thick;  allowing 
for  12  windows,  each  8  ft.  high  and  3  ft.  wide,  and  6  doors, 
each  7  ft.  8  in.  high  and  3  ft.  3  in.  wide? 

LUMBER   MEASURE. 

Lumber  is  commonly  estimated  by  board  measure;  1  foot 
being  1  square  foot  of  surface  and  1  inch  in  thickness. 

Less  than  1  inch  in  thickness  counts  the  same  as  1  inch. 
Above  1  inch  the  thickness  is  counted  by  fourths;  as,  li, 
1^,  If,  etc.     One  cubic  foot  counts  as  12  ft.  of  lumber. 

The  width  of  a  tapering  board  is  half  the  sum  of  the  end 
widths. 

7.  Find  contents  and  cost  of  a  board  14  ft.  long,  1  ft.  4  in. 
wide,  at  1^  cents  a  ft. 


ARITHMETIC.  139 

8.  Find  the  contents  of  a  tapering  board  15  ft.  long,  16 
in.  wide  at  one  end  and  11  in.  wide  at  the  other. 

9.  How  many  ft.  in  a  stick  of  timber  30  ft.  6  in.  long 
and  8  in.  square? 

10.  Find  the  cost  of  40  boards  14  ft.  long,  11  in.  wdde,  at 
$32.50  per  thousand. 

11.  Find  cost  of  9  planks  12  ft.  long,  14  in.  wide,  and  3 
in.  thick,  at  $40  per  thousand. 

12.  How  many  ft.  in  45  2-by-4  scantlings  18  ft.  long? 

13.  What  will  328  inch  boards,  12  ft.  long,  8  in.  wide, 
cost  at  $24  per  thousand? 

14.  How  many  ft.  in  a  board  12  ft.  long,  8  in.  wide,  |  in. 
thick? 

15.  In  a  stick  of  lO-by-12  in.  timber  24  ft.  long,  how 
many  ft.? 

16.  Find  the  number  of  ft.  in  8  3-in.  planks  14  ft.  long 
and  10  in.  wide. 

17.  In  a  stick  of  timber  50  ft.  long  12  in.  square,  how 
many  ft.  ?  " 

18.  In  10  4-by-6  in.  joists  18  ft.  long,  how  many  ft.? 

19.  In  8  2-in.  planks  16  in.  wide  18  ft.  long,  how  many  ft.  ? 

20.  In  a  tapering  board  11  ft.  long  18  in.  wide  at  one  end, 
11  in.  wide  at  the  other,  and  I  in.  thick,  how  many  ft.? 

21.  In  2  sticks  of  timber  each  15  in.  square  and  19  ft. 
long,  how  many  ft.  ? 


LIQUID  MEASUEE. 

TABLE. 

2    pints  (pt.)  =  l  quart  (qt.). 
4    qt.  =1  gallon  (gal.). 

3Hgal.  =1  barrel  (bbl.). 

The  gallon  ^=231  cubic  inches.    The  barrel  is  a  measure 
in  estimating  the  capacity  of  tanks  and  cisterns.    Casks  of 


140  CALIFORNIA   SERIES. 

all  sizes  are  used  for  wine,  beer,  oil,  etc.,  the  number  of  gal- 
lons in  each  being  marked  on  the  outside. 

EXERCISE   186. 

1.  How  many  pt.  in  3  qt.  ?     In  3  qt.  1  pt.  ? 

2.  How  many  qt.  in  5  gal.?     How  many  pt.  in  2^  gal.? 

3.  How  many  gal.  in  48  qt?     In  96  qt.? 

4.  What  part  of  a  gallon  is  1  pt.? 

5.  What  part  of  3  qt.  are  3  pt.? 

6.  If  milk  is  worth  8  cents  a  qt.,  what  will  5  gal.  cost? 

7.  How  many  pt.  of  lemonade  can  be  made  from  the 
water  in  a  6-gal.  olla  which  lacks  3  qt.  of  being  full  ? 

8.  How  many  pt.  bottles  can  a  druggist  fill  from  7  gal.  of 
alcohol  ? 

9.  How  much  linseed  oil  can  you  buy  for  $5,  at  12|  ct. 
per  qt.  ? 

10.  If  a  gallon  of  molasses  cost  $.60,  what  will  3  pt.  cost? 

11.  What  will  4  gal.  of  milk  cost  at  2  ct.  per  -i  pt.  ? 

12.  A  physician  uses  1  pt.  of  distilled  water  in  1  day; 
how  long  will  2^  gal.  last  him? 

13.  How  many  pt.  in  1  bbl.  2  qt.  1  pt.? 

14.  How  many  qt.  in  5  bbl.  6  qt.? 

15.  How  many  pt.  in  a  bbl.? 

16.  Reduce  5  bbl.  2  qt.  1  pt.  to  pt. 

17.  Reduce  2  gal.  1  qt.  1  pt.  to  pt. 

18.  Change  ^  bbl.  3  qt.  to  qt. 

19.  Change  1  bbl.  i  gal.  1  pt.  to  pt. 

20.  How  many  bbl.  in  7856  qt.  ? 

21.  Change  9563  pt.  to  higher  denominations. 

22.  Change  9543  qt.  to  higher  denominations. 

23.  Change  86543  pt.  to  higher  denominations. 

24.  Reduce  6754  gal.  to  higher  denominations. 

25.  A  maker  of  patent  medicine  has  16  gal.  prepared; 
how  many  pt.  bottles  does  he  need? 

26.  Find  the  cost  of  2  bbl.  4  gal.  2  qt.  1  pt.  of  vinegar  at 
5  ct.  a  pt. 


ARITHMETIC. 


141 


27.  What  will  be  the  cost  of  the  following  lots  of  molas- 
ses at  $.75  per  gal.:  41  gal.  3  qt.  1^  pi,  25  gal.  7  qt.  1  pt., 
and  9  gal.  3  qt.  1^  pt.? 

28.  A  man  sells  in  one  week  73  gal.  3  qt.  of  oil,  in  an- 
other 60  gal.  2  qt.,  in  another  40  gal.  1  qt.,  and  in  a  fourth 
65  gal.  2  qt.;  what  is  it  worth  at  $.17  a  gal.? 

29.  A  Ventura  oil  well  yields  150000  gal.  of  refined  oil; 
how  many  5-gal.  cans  will  it  fill,  and  what  is  the  value  at 
$1.75  a  can? 

30.  4  gal.  1  pt.  is  what  fraction  of  a  bbl.  ? 

31.  What  is  the  value  of  .75  bbl.? 

32.  13  gal.  1  pt.  is  what  fraction  of  |  of  a  bbl.? 

33.  Reduce  ^  of  16  gal.  2  qt.  to  the  fraction  of  1-|  bbl. 

34.  Reduce  |  gal.  to  the  decimal  of  a  bbl. 

35.  How  many  quarts  in  .375  of  a  bbl.? 


DRY  AND  LKiUlD  MEASURE— METRIC  SYSTEM. 

The  unit,  or  liter,  is  the  cubic  decimeter=1.057  quarts. 


Ph 

%^ 

,  • 

■4-S 

t-' 

CD 

r-{ 

CD 

s 

*  '— 1 

O 

.1—1 

-M 

Qi 

-f^ 

■+^ 

-(-^ 

•  1— 1 

c3 

•  ^H 

o 

rl4 

1     1 

•  r— ( 

1     1 

1—1 

2 

.1—1 

•  1— 1 

.1— ( 

.  I—H 

ri 

•  1— 1 

O 

o 

-t-i 

a 

o 

^  >> 

-^ 

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-r=i 

^ 

O 

r^ 

EXERCISE   187. 

1.  In  a  cubic  meter  of  water 
are  how  many  liters? 

2.  Read  9852.436  liters  with 
any   one  of  the  names  in  the 

0     000     0.000     table  as  the  unit. 

3.  How  many  liters  are  there  in  a  cubic  dekameter  of 
water? 

4.  A  rectangular  cistern  is  2.75  meters  long.  1.82  meters 
wide,  and  1.12  meters  high,  inside  measurement.  How 
many  liters  of  water  will  it  hold?     How  many  kilograms? 

5.  Find  the  answer  to  the  last  example  in  gallons,  and 
in  pounds  avoirdupois. 


142  CALIFORNIA    SERIES. 


WEIGHT. 

This  is  the  measure  of  the  force  that  draws  all  bodies 
downward. 

The  table  in  common  use  is  that  of 

AVOIRDUPOIS  WEIGHT. 

16  ounces  (oz.)  =  l  pound  (ft.)- 
100  1b.  =1  cental. 

20  centals  =lton(T.). 

The  cental  is  also  called  the  quintal  and  hundredweight. 
100  lb.  of  grain  or  flour  is  called  a  cental ;  100  ft),  dried  fish, 
a  quintal ;  100  lb.  of  other  coarse  substances,  a  hundred- 
weight (cwt.). 

EXERCISE   188. 

1.  How  many  oz.  in  3  lb.?     In  4  lb. ? 

2.  In  5  cwt.  25  lb.  how  many  lb.  ? 

3.  How  many  lb.  in  4^  centals? 

4.  What  is  a  quintal  of  fish  worth  at  6  ct.  per  lb.? 

5.  How  many  lb.  in  2  T.  ?     In  3^  T.? 

6.  What  part  of  a  ton  are  400  ft).?     500  ft).? 

7.  What  part  of  a  lb.  are  4  oz.?     12  oz.? 

8.  Wliat  part  of  a  cwt.  are  75  ft).?     66|  lb.? 

9.  20  lb.  are  what  part  of  a  cental  ? 

10.  60  lb.  are  what  part  of  a  cental? 

11.  At  5  ct.  per  lb.  what  are  7  cwt.  of  beef  worth? 

12.  At  $.12^  per  lb.  what  are  12  lb.  of  cheese  worth? 

13.  How  much  are  2^  centals  of  wheat  worth  at  $.94? 

14.  A  druggist  buys  powdered  bloodroot  at  $1.00  per  ft). 
and  sells  it  for  12-|  ct.  per  oz.;  what  does  he  gain? 

15.  A  man  buys  240  lb.  of  sugar  at  the  rate  of  12  lb.  for 
a  dollar,  and  pays  for  it  in  peaches  at  2  ct.  a  lb.;  how 
many  lb.  of  peaches  will  it  take? 

16.  A  woman  takes  300  lb.  of  honey  from  her  hives  each 
month.     What  is  it  worth  for  one  year  at  $5.00  per  cwt.  ? 


ARITHMETIC.  143 

TROY  WEIGHT. 

The  following  table  is  used  for  weighing  gold,  silver,  and 
precious  stones : 

24  grains  (gr.)=l  pennyweight  (pwt.). 
20  pwt.  =1  ounce  (oz.). 

12  oz.  =1  pound  (ft).). 

The  Avoirdupois  pound=7000  Troy  grains. 

The  purity  of  gold  and  silver  was  formerly  expressed  in 
24ths,  or  carats;  thus  if  yf  of  a  piece  of  metal  was  gold,  it 
was  18  carats  fine. 

This  reckoning  of  fineness  of  silver  and  gold  by  24ths,  or 
carats,  is  now  used  only  for  jewelry.  In  buying  and  sell- 
ing, as  well  as  in  coining,  fineness  is  now  reckoned  in  thou- 
sandths; thus, 

925  fine  means  that  -fw^  of  the  entire  weight  is  pure  metal. 

EXERCISE   189.    (Written.) 

1.  How  many  gr.  heavier  is  the  Avoirdupois  lb.  than  the 
Troy  ft).? 

2.  Which  is  the  heavier  and  by  how  many  gr.,  the 
Avoirdupois  oz.  or  the  Troy  oz.? 

3.  In  1^  pwt.  how  many  gr. ?     In  3  pwt.? 

4.  How  many  gr.  and  pwt.  in  80  gr.  ? 

5.  What  part  of  a  pwt.  are  12  gr.?     3  gr.?     8  gr.? 

6.  How  many  pwt.  in  3  oz.?     In  5  oz.?     In  7^  oz.? 

7.  In  70  pwt.  how  many  oz.?     In  90  pwt.? 

8.  In  72  oz.  how  many  lb.  ?     In  84  oz.  ? 

9.  How  many  oz.  in  9  lb. ?     In  6  lb.?     In  7^  lb.? 

10.  What  part  of  a  lb.  are  4  oz.  ?     6  oz.? 

11.  What  part  of  a  ft),  are  40  pwt.?     60  pwt.? 

12.  If  a  pwt.  of  gold  is  worth  $.95,  what  are  12  pwt. 
worth?     What  are  12  gr.  worth? 

13.  If  a  salt  spoon  weighs  5  pwt.,  how  many  spoons  can 
be  made  from  2  ft),  of  silver? 

14.  Reduce  35624  avoirdupois  oz.  to  cwt. 


144  CALIFORNIA   SERIES. 

15.  Change  16256  avoirdupois  oz.  to  higher  denomina- 
tions. 

16.  How  many  cwt.  in  40607  avoirdupois  oz.  ? 

17.  In  267235  ib.  how  many  T.? 

18.  Change  8420724  avoirdupois  oz.  to  T. 

19.  Reduce  24  ft).  9  oz.  6  gr.  Troy,  to  gr. 

20.  Change  855  gr.  to  higher  denominations. 

21.  Change  25  ft).  7  oz.  18  pwt.  9  gr.  to  gr. 

22.  Reduce  6  ft).  8  oz.  6  pwt.  to  gr. 

23.  Reduce  3756  ib.  to  gr. 

24.  Reduce  217  T.  35  ft),  to  ft). 

25.  Change  7  T.  9  cwt.  18  ft),  to  ft). 

26.  If  a  man  has  987567  ft),  of  wheat  how  many  centals 
has  he? 

27.  How  many  quintals  of  fish  in  9875  ft).  ? 

28.  What  are  78569  ft),  of  wheat  worth  at  $.95  a  cental? 

29.  Find  the  value  of  8564  oz.  of  tea  at  $1.25  per  ft). 

30.  Change  97546  gr.  Troy  to  higher  denominations. 

31.  How  many  Troy  ft).,  oz.,  and  gr.  in  85643  gr.? 

32.  What  are  755  centals  of  barley  worth  at  $.84^  a 
cental ? 

33.  Find  the  cost  of  896  lb.  of  dried  apricots  at  $25  per 
cwt. 

34.  What  are  745  ft),  of  dried  ginseng  worth  at  $.12^  per 
oz.? 

35.  A  lady  buys  table  silver  weighing  2  lb.  3  oz.  10  pwt. 
at  the  rate  of  $1.60  per  oz.     Find  the  cost. 

36.  The  Monitor  Co.  ships  by  the  Wells-Fargo  Express 
Co.  9  bars  of  silver  bullion,  each  bar  weighing  1000  oz.,  the 
whole  valued  at  $9612;  what  is  the  value  of  an  oz.? 

37.  A  car-load  of  ore  taken  to  Denver  yields  800  oz.  of 
silver  to  the  ton;  if  there  are  10  tons  in  the  car,  what  is  it 
worth  according  to  the  vahiation  in  the  preceding  example  ? 

38.  A  gold  mine  was  sold  for  $200000;  if  the  ore  yields 
$248  a  load,  how  many  loads  will  it  take  to  pay  for  it,  and 


ARITHMETIC. 


145 


how  many  bricks  will  it  make  of  500  oz.  each,  if  an  oz.  of 
gold  is  worth  $20? 

39.  Reduce  5  oz.  5  pwt.  to  the  fraction  of  a  ib. 

40.  Express  as  a  decimal  -^-^  of  y^o"  ^• 

41.  Express  .08  ib.  as  units  of  lower  denominations. 

42.  What  decimal  of  a  lb.  is  .24  oz.? 

43.  What  decimal  of  a  lb.  is  4  oz.  10  pwt.? 

44.  Change  2  centals  15  oz.  to  the  fraction  of  a  T. 

45.  Change  3  centals  8  oz.  to  the  decimal  of  a  T. 

46.  Express  in  lower  denominations  yj  of  a  T. 

47.  Express  in  lower  denominations  .075  of  a  T. 
4S. 


1  T 

•2    ^  • 


j\  of  a  T.  is  what  fraction  of  |  of  2 

49.  What  fraction  of  3  T.  is  .065  of  a  T.? 

50.  Reduce  17  centals  50  ib.  to  the  decimal  of  a  T. 


WEIGHT— METRIC  SYSTEM. 

The  gram,  or  unit,  is  the  weight  of  a  cubic  centimeter  of 
pure  water. 

The   gram=15.4   grains 

Troy  weight.    It  is  the  unit 

.rp,  for    very    small    weights. 

For  common  purposes  the 


c3 


tJO 


c5 


0000000.000    kilogram  is  the  unit 

The  kilogram=2.2  pounds  avoirdupois. 
The  ton  =2204.6  "  '' 


EXERCISE   190. 

1.  2784.683  grams:  read  this  value  as  kilograms;  as  hek- 
tograms;  as  decigrams. 

2.  What  is  the  weight  of  a  liter  of  water? 

3.  434.28  grams:  give  this  value  in  Troy  wt.;  in  Avoir- 
dupois wt. 

10— A 


146 


CALIFORNIA   SERIES. 


4.  A  cistern  holds  74625837  grams  of  water,  how  many 
liters  of  water  does  it  contain  ? 

5.  How  many  liters  in  a  barrel  of  water? 

6.  A  garden  is  115  meters  long,  87.5  meters  wide;  when 
rain  falls  on  it  to  the  depth  of  1^  centimeters,  how  many 
liters  of  water  has  it  received  ? 


CIEGULAB  MEASURE. 


A  circle  is  a  flat  surface  whose  edge  is  a  uniformly  curved 
line. 

The  edge,  or  circumference^  is  divided  into  360  equal  parts 

called  degrees.    Learn  the  names  connected  with  the  circle 

from  the  figure. 

TABLE. 

60  seconds  (")  =  1  minute  (')• 
60'  =  1  degree  (°). 

90°  =  1  quadrant. 

4  quadrants     =  1  circumference. 


ARITHMETIC.  147 

EXERCISE   191.    (Written.) 

1.  What  part  of  a  circumference  is  90°?     What  part  is 
180"? 

2.  How  many  seconds  in  2°  1'  5"? 

3.  How  many  degrees  in  a  quadrant? 

4.  How  many  degrees  in  \  a  circumference? 

5.  How  many  degrees  in  3600"? 

6.  In  2^  minutes,  how  many  seconds? 

7.  How  many  degrees  in  -^  of  a  quadrant?     In  3-  of  a 
quadrant?     In  |  of  a  quadrant? 

8.  2\  quadrants  are  what  part  of  a  circumference  ? 

9.  How  many  seconds  in  29°  35'  26"? 

10.  Change  943765"  to  higher  denominations. 

11.  What  is  I  of  45°? 

12.  What  part  of  a  quadrant  is  25"? 

13.  Reduce  .05  of  a  circumference  to  the  fraction  of  3  cir- 
cumferences. 

14.  Change  .32  of  a  quadrant  to  the  decimal  of  a  circum- 
ference. 

15.  What  part  of  5°  2'  3"  is  1°  40'  41"? 

16.  Reduce  9°  25'  48"  to  the  decimal  of  25°  3'  28". 

17.  What  fraction  of  9°  8'  is  \  of  22°  50'? 

18.  Change  .125  of  a  degree  to  minutes  and  sec. 


TIME. 


The  earth  is  a  huge  pendulum  marking  time  by  its 
motion;  turning  once  around  on  its  axis  marks  one  day. 

TABLE. 

60  seconds  (sec.)  =  1  minute  (min.). 
60  min.  =1  hour  (hr.). 

24  hr.  =1  day  (da.). 

7  da.  =1  week  (wk.). 

365  da.  =  1  common  year. 

366  da.  =  1  leap  year. 


148  CALIFORNIA   SERIES. 

All  years  divisible  by  4  are  leap  years. 

Exception:  those  divisible  by  100  and  not  by  400  are 
common  years. 

The  year  has  12  months  of  28,  29,  30,  or  31  days.  The 
following  rhymes  serve  to  keep  the  lengths  of  the  months 
in  the  memory: 

Thirty  days  hath  September, 
April,  June,  and  November, 
February  twenty-eight, 
Thirty-one  the  others  rate. 

The  extra  day  for  leap  year  is  added  to  February,  mak- 
ing 29. 

For  school  purposes  and  in  working  examples,  4  weeks 
are  counted  as  1  month. 

In  reckoning  interest  30  days  are  counted  as  1  month. 

EXERCISE    192.    (Written.) 

1.  How  many  min.  in  |  of  an  hour?     In  ^  of  an  hour? 
In  I  of  an  hour? 

2.  Howmany  sec.inf  of  amin.?  f  of  a  min.?  f  of  a  min.? 

3.  How  many  min.  in  5  hr.?     How  many  hr.  in  25  da.? 

4.  Change  3  min.  25  sec.  to  sec.     Change  2  wk.  5  da.  to 
da.;  4  da.  to  hr.;  7  wk.  3  da.  to  da. 

5.  How  many  wk.  in  497  da.?     In  427  da.? 

6.  How  many  da.  in  72  hr.  ?     In  96  hr.  ? 

7.  How  long  is  it  from  25  min.  past  5  a.  m.  to  noon? 

8.  How  many  days  from  Mar.  16  to  June  11  of  the 
same  year? 

9.  How  many  mo.  from  July  4  to  Dec.  4  of  the  same 
year  ? 

10.  Wliat  is  ^  of  1  da.  2  hr.  40  min.  20  sec? 

11.  If  a  horse  trots  1  mi.  in  2  min.  35  sec,  how  long  will 
it  take  him  to  go  3  mi.  at  the  same  rate? 

12.  If  a  man  earns  $3  per  day  and  pays  $5  a  week  for 
board  and  other  expenses,  what  can  he  save  in  6  months  ? 


ARITHMETIC.      '  149 

13.  Reduce  5  hr.  15  min.  25  sec.  to  sec. 

14.  Reduce  2  yr.  11  da.  12  min.  to  min. 

15.  Reduce  3  yr.  37  da.  16  hr.  24  min.  13  sec.  to  sec. 

16.  Reduce  58967379  sec.  to  higher  denominations. 

17.  Change  47675  min.  to  higher  denominations. 

18.  Change  427329  sec.  to  higher  denominations. 

19.  Change  157540  min.  to  higher  denominations. 

20.  Reduce  to  higher  denominations  8567983  sec. 

21.  How  many  min.  are  there  from  25  min.  past  9  p.  m. 
to  15  min.  past  6  the  next  morning? 

22.  How  much  time  is  there  from  9  min.  25  sec.  past  3 
p.  M.  to  8  min.  16  sec.  of  5  a.  m.  of  the  next  day? 

23.  A  young  lady  reads  German  25  minutes  each  day  for 
6  years;  hov/  much  time  does  she  spend? 

24.  If  it  takes  you  25  min.  15  sec.  to  walk  to  school,  how 
much  time  will  you  spend  in  6  mo.  if  you  make  two  trips 
a  day? 

25.  Reduce  f  of  a  common  yr.  to  da. 

26.  What  part  of  a  day  are  2  hr.  30  min.  45  sec.  ? 

27.  What  part  of  6  da.  15  hr.  40  min.  36  sec.  are  3  da.  7 
hr.  50  min.  18  sec? 

28.  Reduce  .075  of  a  da.;  .625  of  a  wk.;  .378  of  a  com- 
mon yr.  to  lower  denominations. 

29.  What  fraction  of  a  wk.  is  2  da.  18  hr.? 

30.  Reduce  .58  of  a  common  yr.  to  lower  denominations. 

31.  Express  4  da.  7  hr.  45  min.  48  sec.  as  a  decimal  of 
34  da.  14  hr.  6  min.  24  sec. 

32.  Find  the  value  of  .975  of  a  yr. 

33.  Express  .125  of  a  yr.  in  lower  denominations. 

34.  22  da.  12  hr.  is  what  part  of  a  mo.? 

35.  2  hr.  40  min.  36  sec.  is  ^vhat  fraction  of  a  mo.? 

36.  Express  11  hr.  33  min.  as  a  fraction  of  a  wk. 

37.  Reduce  31  min.  30  sec.  to  the  fraction  of  a  da. 

38.  Express  in  integers  4.655  yr. 

39.  Express  in  decimals  of  a  week,  3  da.  3  hr. 


150  CALIFORNIA   SERIES. 

40.  2  mo.  3  da.  4  hr.  28  min.  28  sec.  is  what  decimal  of 
6  mo.  1  wk.  2  da.  13  hr.  25  min.  ? 

41.  3  hr.  37  min.  1  sec.  is  what  decimal  of  12  da.  1  hr.  ? 


LONGITUDE  AND  TIME. 

The  earth  turns  on  its  axis  360°,  or  once  around,  in  24 
hours;  therefore  in  one  hour  it  turns  -2V  of  360°=  15°. 

If  then  we  divide  the  degrees  of  longitude  between  two 
places  by  15,  the  quotient  is  the  difference  of  time  in  hours. 
Or  if  we  multiply  the  difference  of  time,  in  hours,  by  15, 
the  product  is  the  difference  in  degrees  of  longitude.  Now 
the  hour  and  degree  have  the  same  division  into  minutes 
and  seconds;  hence, 

Hoursxl5  =  degrees  of  longitude. 

Minutes  of  timexl5  =  minutes  of  longitude. 

Seconds  of  timexl5  =  seconds  of  longitude. 

And  difference  of  longitude  divided  by  15  gives  corre- 
sponding divisions  of  difference  of  time. 

EXERCISE   193.    (Written.) 

1.  What  is  the  difference  of  longitude  of  two  places  whose 
difference  of  time  is  3  hr.  25  min.  30  sec? 

2.  Two  towns  have  a  difference  of  longitude  of  17°  48' 
36";  what  is  the  difference  of  time? 

In  changing  difference  of  time  to  difference  of  longitude, 
we  multiply  by  15  and  divide  the  products  of  the  minutes 
and  seconds  by  60  to  reduce.  Multiplying  by  15  and  divid- 
ing by  60  is  equivalent  to  dividing  by  4,  hence  we  may 
shorten  the  work  in  this  Avay: 

1.  3'^'"-  2  5'"^'^-  30^*^"- 

15 


4  5°  15'  3  0" 

25h-4=_6^         3  0~4:=    7 

51°  2  2'  ^ws.  51°  22' 30". 


ARITHMETIC.  151 

2.   By  the  opposite  process  longitude  is  changed  to  time. 

15)17° 48' ar 

-|  hr.  Omin.  O  2.sec. 

2X4=_8  3X4::^^!  2 

-^  -j^min.  14-2-  *cc. 


Ans.  r-^- 11"^^"- 14-1 


sec. 


The  student,  with  a  Httle  practice  in  this  method,  will 
work  any  example  without  using  his  pencil  except  to  write 
the  answer. 

3.  Two  hours  difference  in  time  corresponds  to  how  many 
degrees  difference  in  longitude? 

4.  What  difference  in  longitude  corresponds  to  \  hr.  dif- 
ference in  time? 

5.  If  the  difference  in  time  between  two  places  is  2  hr. 
15  min.,  what  is  the  difference  in  longitude? 

6.  When  it  is  noon  at  Sacramento,  what  time  is  it  15° 
east  of  that  place? 

7.  A  meteor  is  observed  by  two  persons  whose  difference 
in  longitude  is  8°  30';  what  will  be  the  difference  in  time 
recorded  ? 

8.  The  difference  in  time  between  two  places  is  2  hr.  25 
min.  6  sec;  what  is  their  difference  in  longitude? 

9.  What  is  the  difference  in  longitude  between  two  places 
whose  difference  in  time  is  1  hr.  24  min.  16  sec? 

10.  When  the  difference  in  time  between  two  places  is 
3  hr.  14  min.  28  sec,  what  is  their  difference  in  longitude? 

11.  Find  the  difference  in  longitude  when  the  difference 
in  time  is  5  hr.  13  min.  12  sec. 

12.  What  is  the  difference  in  longitude  when  the  differ- 
ence in  time  is  4  hr.  8  min.  12  sec? 

13.  When  the  difference  in  time  is  17  hr.  9  min.  14  sec, 
what  is  the  difference  in  longitude? 

14.  What  is  the  difference  in  longitude  of  two  places 
whose  difference  in  time  is  15  hr.  14  min.  13  sec? 


152  CALIFORNIA   SERIES. 

Counting  from  the  meridian  that  passes   through  the 
observatory  of  Greenwich,  near  London,  the  longitude  of 

New  York  is  74°  3"  W.  Paris,  2°  20'  22"  E. 

New  Orleans,  90°  5'  W.  Boston,  71°  3'  30"  W. 

Berlin,  13°  23'  53"  E.  Pekin,  116°  28'  54"  E. 

Chicago,  87°  37'  30"  W.  Montreal,  73°  34'  W.  ^ 

Cincinnati,  84°  26'  W.  St.  Petersburg,  30°  18'  E.  | 

St.  Louis,  90°  15'  16"  W.  St.  Paul,  93°  5'  W. 

Bombay,  72°  53'  E.  San  Francisco,  122°  24'  15"  W. 

Mexico,  99°  5'  AV.  Omaha,  95°  56'  AV. 

Washington,  77°  2'  48"  W.  Los  Angeles,  118°  18'  W. 

Albany,  73°  32'  W. 

15.  What  is  the  difference  in  time  between  San  Francisco 
and  St.  Paul? 

16.  Find  the  difference  in  time  between  New  York  and 
New  Orleans. 

17.  What  is  the  difference  in  time  between  Berlin  and 
Bombay  ? 

18.  Find  the  difference  in  time  between  San  Francisco 
and  New  York. 

19.  What  is  the  difference  in  time  between  Chicago  and 
St.  Petersburg? 

20.  Find  the  difference  in  time  between  the  City  of  Mex- 
ico and  St.  Louis. 

21.  What  is  the  difference  in  time  between  Cincinnati 
and  Washington,  D. C? 

22.  What  is  the  difference  in  time  between  Pekin  and 
Montreal  ? 

23.  When  it  is  noon  at  San  Francisco,  what  time  is  it  at 
Paris?  I 

24.  When  it  is  six  o'clock  p.  m.  in  Boston,  what  time  is 
it  at  Pekin? 

25.  When  it  is  midnight  in  Paris,  what  time  is  it  in  St. 
Petersburg  ? 

i 

As  the  earth  turns  eastward  in  its  daily  motion,  the  sun     ' 


ARITHMETIC.  153 

appears  to  move  westward  15°  per  hour.  When  it  is  noon 
at  any  point,  it  will  be  past  noon  at  all  places  east  of  that 
meridian,  and  before  noon  at  all  places  west;  therefore  we 
reckon  to  the  west  for  earlier  time,  and  to  the  east  for  later. 

26.  When  it  is  10^  o'clock  a.  m.  at  Sacramento,  longitude 
121°  26'  W.,  what  is  the  longitude  of  the  places  having  the 
following  times:  7  hr.  20  min.  a.  m.:  2  lir.  25  niin.  p.  m.;  1. 
lir.  10  min.  p.  M. ;  5  hr.  15  min.  a.  m.? 

27.  What  is  the  longitude  of  a  place  where  it  is  3  hr.  30 
min.  p.  M.  when  it  is  7  hr.  30  min.  A.  m.  in  Albany? 

28.  At  Paris  it  is  4  p.  ]\r.,  and  at  the  same  time  it  is  2  a.  m. 
of  the  next  day  in  another  place;  give  the  longitude  of  the 
last  place. 

29.  In  Boston  it  is  1:25  p.  m.  when  it  is  1.25  a.  m.  of  the 
next  day  at  another  place;  what  is  the  longitude  of  that 
place? 

30.  What  is  the  longitude  of  the  place  where  it  is  9  hr. 
25  min.  a.  m.  when  it  is  6  hr.  30  min.  p.  m.  of  the  same  day 
at  Paris? 

31.  What  is  the  longitude  of  a  place  where  it  is  15  min. 
past  8  o'clock  a.  m.,  when  it  is  7  hr.  30  min.  a.  m.  in  Omaha? 

32.  When  it  is  45  min.  past  11  p.  m.  in  Montreal,  it  Js  15 
min.  past  2  o'clock  a.  m.  next  day  at  another  place:  what 
is  the  longitude  of  the  second  place? 

33.  What  place  in  the  above  list  has  5  hr.  5  min.  21|  sec. 
earlier  time  than  Paris? 

34.  A  gentleman  travels  from  Boston  to  Springfield  where 
he  finds  that,  by  his  watch,  the  sun  rises  6  min.  6^  sec. 
later  than  in  Boston;  what  is  the  longitude  of  Springfield? 

35.  At  how  late  an  hour  may  news  be  telegraphed  from 
New  York  and  reach  San  Francisco  at  3  a.  m.  ? 

36.  The  absolute  difference  in  time  between  the  Bermu- 
das (which  are  east  of  New  York)  and  New  York  is  37 
min.:  find  the  longitude  of  the  Bermudas. 

37.  Find  the  longitude  of  the  place  where  it  is  20  min. 


154  CALIFORNIA   SERIES. 

past  5  p.  M.  when  it  is  25  min.  past  11  p.  m.  in  St.  Peters- 
burg. 

The  change  of  time  from  place  to  place  is  very  incon- 
venient for  travelers  and  railways.  To  do  away  with  this 
the  United  States  has  been  divided  into  four  districts  by 
lines  running  nearly  north  and  south.  The  eastern  district 
takes  the  time  of  the  75th  degree  of  longitude  west  from 
Greenwich;  the  next  district  takes  the  90th  degree;  the 
next  the  105th,  and  the  western,  which  includes  all  the 
Pacific  states  and  territories,  takes  the  120th  degree. 

In  traveling  across  the  continent  the  minute  hand  of  an 
accurate  watch  keeps  always  right;  the  hour  hand  alone 
needs  changing. 

In  changing  from  local  time  to  the  time  now  kept,  what 
change  was  made  by  San  Francisco,  Ion.  122°  26'  4"  W.? 
What  by  Los  Angeles,  118°  18'?  By  Sacramento,  121°  26'? 
By  San  Diego,  117°  10' 40"? 

Note. — The  examples  in  this  book  are  wrought  for  true  time  and 
not  for  the  standard  time  given  above. 


EEFEREI^CE  TABLES. 

LENGTH. 

7.92  in.  =11. 

6  ft.  =1  fathom. 

120  fathoms  =1  cable-length. 

1  nautical  mile  =1.153  common  mile. 

1  deg.  at  equator  =60  nautical  miles  =  69.16  common  miles. 

40  rd.  =  1  furlong. 

4  in.  =1  hand. 

9  in.  =1  span. 

18  in.  =1  cubit. 

12  lines  =1  in. 

33.39  in.  =  1  vara. 


ARITHMETIC.  155 

The  table  of  cloth  measure  seems  to  be  entirely  obsolete. 

SQUARE  OR  SURFACE  MEASURE. 

40  sq.  rd.  =  1  rood. 
100  sq.  ft.  =1  square. 

CUBIC  OR  SOLID  MEASURE. 

16      cu.  ft.  =1  cord  foot  (cd.  ft.). 

8      cd.  ft.  =lcord(cd.). 

16K  cu,  ft.  of  masonry  =  1  perch. 
50  cubic  feet  of  hewn  timber,  or  round  timber  enough  to  make 
40  ft.  of  hewn  timber  =1  ton. 

-  Masonry  is  usually  estimated  by  the  cubic  foot;  but  the 
perch  is  not  unfrequently  used,  and  may  be  that  given 
above,  or  the  old  measure  of  24|  cubic  feet. 

Custom  in  California  has  reduced  the  cord  of  wood  to  an 
uncertain  quantity,  usually  called  96  cubic  feet  or  3  tiers 
of  12-inch  stove  wood,  8  ft.  long  and  4  ft.  high. 

LIQUID  MEASURE. 

4  gills  (gi.)  =  l  pt. 
42  gal.  =1  tierce. 

2  bbl.  =1  hogshead  (hhd.). 

2  hhd.  =  1  pipe  or  butt. 

2  pipes        =1  tun. 

BEER  MEASURE. 

2      pt.    =lqt. 
4      qt.    =1  gal. 
36      gal.  =  1  bbl. 
l>^bbL  =  lhhd. 

The  beer  gallon^282  cubic  inches. 

DRY  MEASURE. 

2  pt.  =1  qt. 

8  qt,  =1  peck  (pk.). 

4  pk.  =  1  bushel  (bu.) 

The  bushel=21 50.42  cubic  inches. 


156 


CALIFORNIA    SERIES. 


The  table  of  dry  measure  is  seldom  or  never  used  in  Cal- 
ifornia, grain  and  vegetables  being  bought  and  sold  by 
weight.  It  is  still  in  common  use  in  the  eastern  states  and 
the  Mississippi  valley,  but  the  bushel  differs  somewhat  in 
the  different  states. 

The  English  standard  bushel=:2218.19  cu.  in. 

APOTHECARIES'  WEIGHT. 

20  gr.        =1  scruple  (scr.) 

3  scr.      =1  dram. 

8  drams  =  1  oz. 
12  oz.        =1  Itx 

Used  in  mixing  medicines  but  not  in  buying  and  selling. 
The  pound,  ounce,  and  grain  are  the  same  as  in  Troy 
weight. 

DIAMOND  WEIGHT. 

16  parts       =  1  carat  gr. 
4  carat  gr.  =  1  carat. 
1  carat       =3.17  gr.  Troy. 

Circular  measure  has  been  expressed  decimally  by  divid- 
ing the  quadrant  into  100  equal  parts  called  grades,  the 
tenths  being  called  minutes,  and  the  hundredths,  seconds. 
Thus,  45°  30'  15"  would  be  50.56  grades,  but  the  French 
people,  who  invented  this,  have  not  generally  adopted  it. 


PAPER  AND  BOOKS. 


24  sheets  =  1  quire 

(qr.). 

20  qr 

=  1  ream 

(rm.). 

2  rm 

I. 

=  1  bundle  (bun.). 

5  bun. 

=  1  bale. 

A  sheet  fol 

ded  into 

2 

leaves  forms  a  folio. 

A     " 

i         (I 

4 

li             a 

a  quarto  or  4to. 

A     '' 

C                   C  i 

8 

(C                   il 

an  octavo  or  8vo. 

A     '' 

I             a 

12 

((               n 

a  duodecimo  or  12mo 

A    " 

t             a 

18 

li            (I 

an  18mo. 

A    " 

:              cc 

36 

11            t( 

a  36mo, 

ARITHMETIC.  157 

ENGLISH  AtONEY. 

4  farthings  (far.)  =  i>?^nny  (d.)- 

12  d.  =1  .slimiu^Js^)._ 

20  s.  =1  pound  (£)." 

The  pound=$4.86. 

FRENCH  MONEY. 

The  franc  is  the  unit,  tenths  being  called  decimes,  and 
hundredths,  centimes.     1  franc;=$.186. 

CALIFORNIA  MEASURES. 

The  following  approximate  measures  may  be  found  of 
practical  value: 

A  cord  of  stovewood  is  a  pile  8  ft.  long,  4  ft.  high,  and 
three  tiers  wide. 

Hay  is  not  dried  grasses,  as  in  most  of  the  states,  but  oats, 
barley,  or  wheat  cut  before  the  grain  is  fully  formed;  a  ton 
of  hay  in  the  stack,  well  settled,  is  a  cube  each  of  whose 
edges  is  8  ft.,  or  8X8X8=512  cu.  ft.  If  loose  on  the  wagon 
it  is  10X10X10=1000  cu.  ft. 

4  cu.  ft.  of  unshelled  corn^^l  cental  of  shelled  corn.  If 
loosely  thrown  in  and  not  settled,  4-|  cu.  ft.=l  cental. 

A  square  whose  edge  is  70  paces^l  acre. 

Flowing  water  is  measured  by  the  inch  in  California. 
The  inch,  however,  is  not,  at  present  (1887),  a  fixed  unit, 
but  varies  by  custom  of  the  different  companies  supplying 
water  for  irrigation  or  domestic  purposes.  By  statute,  the 
inch  is  the  water  flowing  through  a  square  inch  of  vertical 
surface,  the  center  of  the  opening  being  A\  inches  below 
the  surface  of  the  reservoir  from  which  the  water  is  flowing. 
The  amount  is  given  as  1.394  cubic  feet  per  minute. 

COUNTING  TABLE.        LONG  TON  TABLE. 

12  units  make  1  dozen  (doz.).  28  ft.  make  1  quarter. 

12doz.       ''       1  gross  (gro.).  112"       "       1  cwt. 

12  gro.       "       1  great  gro.  20  cwt.  *'       IT. 
20  units     *'      1  score  (sc). 


158  CALIFORNIA   SERIES. 


ADDITION. 

EXERCISE  194.   (Written.) 

1.  Add  69  rd.  2^  yd.,  1  mi.  14  rd.  2  yd.  2  ft.  3  in.,  16  rd. 
9  in.,  and  25  rd.  11  ft. 

2.  Add  7  yd.  2  ft.,  5  yd.  1^  ft.,  2  ft.  9^  in.,  3  yd.  1  ft.  6^ 
in.,  2|  ft.,  and  4^  yd. 

3.  Add  25  yd.  1  ft.  9  in.,  32  yd.  1  ft.  8  in.,  35  yd.  6  ft.  4 
in.,  7  yd.  2  ft.  11  in.,  and  9  ft. 

4.  Add  23  mi.  118  rd.  14  ft.,  19  mi.  137  rd.  11  ft.,  8  mi. 
62  ft.  8  in.,  23  mi.  147  rd.  6  in.,  and  9  rd.  7  in. 

5.  Add  22  rd.  2  yd.  2  ft.,  18  rd.  4  yd.  2  ft.,  22  rd.  6  yd. 
1  ft.,  and  16  rd.  4  ft.  3  in. 

6.  Add  7  mi.  59  rd.  6  ft.  7  in.,  8  mi.  96  rd.  7  ft.  8  in.,  5 
mi.  9  rd.  8  in.,  26  mi.  87  rd.  8  ft.  3  in. 

7.  Add  71  mi.  23  rd.  4^  yd.,  9  mi.  17  rd.  2  yd.  2^  ft.,  23 
mi.  3  yd.  9  in. 

8.  Add  1^  yd.  3  in.,  2  ft.  4  in.,  and  3^  ft. 

9.  Add  i  yd.,  ^  ft,  and  i  rd. 

10.  Add  I  mi.,  ^  rd.,  ^  yd.,  and  |  ft. 

11.  Add  79  ch.  3  rd.  16  1.,  65  ch.  2  rd.  11  1.,  33  ch.  2  rd. 
6  l.,46ch.  1  rd.  13  1.,  75  ch  2  1. 

12.  Add  75  A.  4  sq.  rd.  9  sq.  yd.  72  sq.  in.,  27  A.  48  sq. 
rd.  18  sq.  yd.  92  sq.  in.,  7  A.  100  sq.  rd.  29  sq.  yd.  8  sq.  ft. 
139  sq.  in.,  and  7  sq.  yd.  129  sq.  in. 

13.  Add  -|  A.  and  f  sq.  yd. 

14.  Add  f  A.,  "I  sq.  rd.,  and  f  of  a  sq.  yd. 

15.  Add  5  cd.  7  cd.  ft.,  2  cd.  2  cd.  ft.  12  cu.  ft.,  6  cd.  ft. 
15  cu.  ft.,  7|  cd.,  3  cd.  2  cu.  ft. 

16.  How  many  cu.  yd.  and  ft.  in  three  bins,  the  first  con- 
taining 95  cu.  yd.  26  cu.  ft.  985  cu.  in.,  the  second,  87  cu. 
yd.  19  cu.  ft.  876  cu.  in.,  the  third,  98  cu.  yd.  3  cu.  ft.  875 
cu.  in.? 

17.  A  man  buys  3  lots  of  vinegar;  the  first  is  29  gal.  2 


ARITHMETIC.  159 

qt.  1  pt.,  the  second,  16  gal.  3  qt.,  the  third  11  qt.  1  pt.; 
how  much  did  he  huy,  and  what  will  it  sell  for  at  10  cents 
a  qt.  ? 

18.  A  man  sold  three  lots  of  beans;  the  first,  5825  pt., 
the  second,  4285  pt.,  the  third,  3426  pt. ;  how  many  bushels 
did  he  sell,  and  what  did  they  amount  to  if  retailed  at  12^ 
ct.  a  qt.? 

19.  A  woman  picked  in  one  day  1  bu.  4  qt.  1  pt.  of  straw- 
berries, the  next  day,  \  bu.  3  qt.  1  pt.,  the  third  day,  27  qt. 

1  pt. ;  how  many  qt.  boxes  can  she  fill,  and  what  will  she 
receive  at  12^  ct.  a  box? 

20.  A  carpenter  worked  for  $.35  an  hour;  on  Monday  he 
worked  9  hr.  15  min.,  Tuesday  8  hr.  20  min.,  Wednesday 
11  hr.,  Thursday  10  hr.  35  min.,  Friday  9  hr.  45  min.,  and 
Saturday  6  hr.  50  min.;  what  did  he  receive  for  his  week's 
work? 

21.  Add  1  doz.  3,  2  gro.,  3  doz.,  1  sc,  3  gro.  5  doz.  4. 

22.  How  many  sheets  in  2  bun.  1  rm.  3  qr.,  3  bun.,  17 
sheets,  1  bun.  1  qr.  ? 

23.  A  traveler  in  England  spends  £6  17s.  5d.  in  one  week, 
the  next,  £7  lis.  4d.,  the  third,  £9  7s.  3d.;  how  much  did 
he  spend? 

24.  A  school  girl  paid  \  dollar  for  paper,  10  cents  for 
pencils,  $1,374  for  a  reader,  $.95  for  an  arithmetic,  and  $.25 
for  a  slate;  what  was  the  entire  cost? 

25.  A  man  sold  4  lots  of  baled  hay;  the  first  weighed  14 
T.  13  cwt.  75  ft).,  the  second,  25  T.  12  cwt.  26  ft).,  the  third, 

2  T.  5  cwt.  14  ft).,  and  the  fourth,  17  T.  16  cwt.  29  ft).;  how 
much  did  it  all  weigh? 

26.  Add  84  T.  12  cwt.  74  ft).  6  oz.,  23  T.  12  cwt.  26 
ft).  8  oz.,  51  T.  16  cwt.  45  ft).  15  oz.,  81  T.  5  cwt.  4  ft). 
7  oz. 

27.  Three  miners  have  the  following  amounts  of  gold 
dust :  the  first,  5  ft).  9  oz.  14  pwt.,  the  second,  3  ft).  7  oz.  13 
pwt.,  the  third,  2  ft).  4  oz.  11  pwt.;  how  much  have  all? 


160  CALIFORNIA   SERIES. 

SUBTEACTIO]^. 

EXERCISE    195.    (Written.) 

1.  A  man  owning  a  farm  of  160  A.  sold  at  one  time  25 
A.  74  sq.  rd.,  at  another  74|  A.,  and  at  another  \  as  much 
as  at  the  first  sale;  how  much  had  he  left? 

2.  Take  3  mi.  110  rd.  4  yd.  2  ft.  from  7  mi.  25  rd.  3  yd.  4  ft. 

3.  Find  the  difference  between  two  fields:  one  is  14  ch. 
43  1.  by  17  ch.  25  1.;  the  other,  8  ch.  11  1.  by  15  ch. 

4.  From  48  cu.  yd.  12  cu.  ft.  1236  cu.  in.  take  28  cu.  yd. 
24  cu.  ft.  1500  cu.  in. 

5.  From  4  gal.  2  qt.  of  syrup  1  gal.  3  qt.  1  pt.  was  drawn; 
what  amount  was  left? 

6.  A  merchant  has  two  barrels  of  kerosene,  one  holding 
Sl-J  gal.,  the  other  30  gal.  1  qt.  He  sold  at  diff'erent  times 
6  gal.  2  qt,  5  gal.  3  qt.,  5-J  gal.,  7f  gal.,  and  28  gal.;  what 
did  his  sales  amount  to  at  27  cents  per  gallon  and  what 
amount  has  he  left? 

7.  A  grocer  buys  at  one  time  7  cwt.  11  oz.  of  tea,  at 
another  6  cwt.  38  lb.  7  oz.  He  sells  11  cwt.  79  lb.  8  oz.; 
what  has  he  left? 

8.  From  .625  Troy  ft.  take  4.25  Troy  oz. 

9.  From  1  cwt.  take  ^  of  ^  of  72  ft.  12  oz. 

10.  A  silver  butter  dish  weighs  1  ft.  2  oz.  5  pwt.,  and  1 
doz.  teaspoons  weigh  11  oz.  17  pwt.  18  gr.;  find  the  differ- 
ence in  weight. 

11.  From  2  ft.  Apothecaries'  weight  take  9  oz.  1  dr.  2 
scr.  7  gr. 

12.  Take  3  yr.  4  da.  3  hr.  from  5  yr.  2  mo.  2  wk.  1  da.  7  hr. 

13.  From  3  mo.  take  2  wk.  4  da.  8  hr.  19  min.  29  sec. 

14.  Find  the  difference  in  days  between  the  first  half  of 
the  year  1885  and  the  time  from  Christmas  to  the  fourth 
of  July,  1884-5. 

Note. — Count  the  day  to  which,  and  omit  the  one  from  which 
you  reckon. 


ARITHMETIC.  161 

15.  Find  the  difference  between  .659  wk.  and  2  wk.  3-J  da. 

16.  A  lady  has  $729  for  house  furnishing.  She  buys  23 
yd.  of  carpeting  at  $1.75  per  yd.,  19  yd.  at  $1.12|  per  yd., 
47  yd.  at  $1.50  per  yd.,  12  yd.  at  $.97;  6  chairs  at  $1.25 
each,  3  at  $2.75  each,  one  for  $16,  and  two  for  $19  each. 
She  spends  \  of  the  remainder  for  hnen  and  silver,  \  of 
what  still  remains  for  kitchen  articles;  Avhat  has  she  left? 

17.  From  the  sum  of  -f-  of  3^  mi.  and  17|  rd.  take  120^  rd. 

18.  How  many  more  seconds  from  New  Years  Day  to  tlie 
Fourth  of  July,  1885,  than  in  the  remainder  of  the  year? 

19.  Find  the  difference  between  3  yr.  17  da.  9  hr.  12  min. 
7  sec.  multiplied  by  4,  and  96  yr.  11  mo.  1  wk.  2  da.  4  hr. 
12  min.  16  sec.  divided  by  3. 

20.  From  |  of  8  T.  16  cwt.  24|  ib.  take  .25  of  a  T. 

21.  From  -fj  of  a  sq.  rd.  take  |  of  a  sq.  yd. 

22.  From  £48  17s.  6d.  2  far.  take  £39  14s.  9d.  3  far. 

23.  The  latitude  of  the  Cape  of  Good  Hope  is  34°  22'  and 
that  of  Cape  Horn  55°  58'  40"  S.;  find  their  difference. 

24.  What  is  the  difference  between  f^  of  a  lb.  and  5  lb. 
4  oz.  8  pwt.? 

25.  Find  the  difference  between  £-|  and  |  of  |s. 

26.  Find  the  difference  in  the  area  of  two  roofs:  one  is 
46  ft.  square,  the  other  contains  46  sq.  ft. 


MULTIPLICATIOIsr. 

EXERCISE  196.    (Written.) 

1.  Multiply  5  mi.  28  rd.  3  yd.  2  ft.  11  in.  by  9. 

2.  Multiply  79  ch.  3  rd.  23  1.  by  7. 

3.  Multiply  158  sq.  rd.  27  sq.  yd.  7  sq.  ft.  138  sq.  in.  by  11, 

4.  MuUiply  98  cd.  13  cu.  ft.  758  cu.  in.  by  13. 

5.  Multiply  5T3bl.  29  gal.  3  qt.  by  23. 

6.  Multiply  7  oz.  17  pwt.  23  gr.  by  96. 

7.  Multiply  75  centals  15  oz.  by  274. 

11— A 


162  CALIFORNIA   SERIES. 

8.  Multiply  9  yr.  7  mo.  3  wk.  5  da.  19  hr.  35  niin.  28  sec. 
by  63. 

9.  What  is  the  length  of  a  fence  inclosing  a  square  field 
one  side  of  which  is  17  rd.  3  yd.  2^  ft.  long? 

10.  If  a  hogshead  of  sugar  weighs  7  cwt.  29  ib.  4  oz., 
what  will  9  hhd.  be  worth  at  9^  ct.  per  pound? 

11.  A  letter  carrier  travels  5  mi.  19  rd.  4  yd.  each  trip; 
how  far  does  he  go  in  the  month  of  January,  mail  being 
delivered  twice  each  day,  four  Sundays  excepted  ? 

12.  A  Avorkman  drinks  a  pint  l)ottle  of  wine  each  day  in 
the  year,  which  costs  him  25  cents  per  bottle;  how  much 
has  he  drunk  in  13  years,  three  of  them  being  leap  years, 
and  what  has  it  cost  him? 

13.  His  wife  buys  a  pint  of  milk  per  day  at  $1.25  per 
month  for  the  same  time;  which  costs  the  most  and  how 
much  ? 

14.  $125  buys  5  A.  24  rd.  19  sq.  yd.  7  sq.  ft.  of  land;  what 
will  $1375  buy? 

15.  A  lady  has  17  silver  spoons,  each  one  weighs  5  pwt. 
6  gr.;  how  much  do  they  all  Aveigh? 


DITISIOK. 

EXERCISE   197.    (Written.) 

1.  Divide  9  mi.  78  rd.  4  yd.  2  ft.  8  in.  by  9. 

2.  Divide  68  ch.  2  rd.  24  1.  by  6. 

3.  Divide  296  sq.  rd.  29  sq.  yd.  8  sq.  ft.  98  sq.  in.  by  16. 

4.  Divide  97  cd.  11  cu.  ft.  979  cu.  in.  by  28. 

5.  Divide  23  bbl.  28  gal.  3  qi  by  19. 

6.  Divide  56  lb.  11  oz.  19  pwt.  21  gr.  by  15. 

7.  Divide  87  cwt.  13  oz.  by  95. 

8.  Divide  24  yr.  11  mo.  2  wk.  3  da.  11  hr.  47  min.  by  17. 

9.  If  294  sacks  of  walnuts  weigh  20600  lb.  what  is  the 
average  weight? 


ARITHMETIC.  163 

10.  If  two  coops  of  fowls  Aveigh  340  ft).  11  oz.  and  there 
are  27  fowls  in  a  coop,  what  is  the  average  weight? 

11.  If  a  township  6  mi.  sq.  he  divided  into  62  equal 
farms,  how  much  land  does  each  contain? 

12.  The  area  of  a  piece  of  land  is  39  sq.  rd.  2  sq.  yd.  6 
sq.  ft.  128  sq.  in.  Its  length  is  11  rd.  2  ft.  8  in.;  what  is  its 
width? 

13.  If  4  men  work  5  days  to  remove  120  cu.  yd.  5  cu.  ft. 
of  earth,  how  much  does  one  man  remove  in  a  day? 

14.  How  many  cups  holding  one  half  pt.  each  can  a  restau- 
rant keeper  fill  from  a  coffee  urn  holding  2  gal.  3  qt.  1  pt.? 

15.  How  many  steel  rails  30  ft.  long  are  needed  to  build 
one  mile  of  railroad  ? 


REVIEW. 


EXERCISE  198.    (Written.) 

1.  A  car  wheel  is  4  ft.  5  in.  in  circumference  and  revolves 
59  times  a  minute;  how  far  does  it  go  in  2  hr.  55  min.  ? 

2.  How  many  cu.  yd.  of  earth  have  been  removed  to 
make  an  irrigating  ditch  1  mi.  8  rd.  long,  3  ft.  wide,  and 
2  ft.  deep. 

2  Metric.  How  many  cu.  yd.  of  earth  have  been  removed 
to  make  an  irrigating  ditch  1649.58  meters  long,  .914  meters 
wide,  and  .609  meters  deep? 

3.  A  man  sells  wheat  at  .$1.50  per  cental  and  receives 
$855.95;  how  much  wheat  has  he  sold? 

4.  If  a  horse  averages  a  mile  in  11  min.  45  sec.  how  far 
does  he  go  in  a  day  of  11  hr.  ? 

4  Metric.  If  a  horse  averages  1609.34  meters  in  11  min. 
45  sec,  how  far  does  he  go  in  11  hr. ? 

5.  How  many  cu.  ft.  in  the  drawers  of  a  school  desk,  one 
of  them  3  ft.  2  in.  long,  2  ft.  10  in.  wide,  and  5  in.  deep,  the 
other  1  ft.  4  in.  wide,  3  ft.  long,  5  in.  deep? 


164  CALIFORNIA   SERIES. 

5  Metric.  Find  the  cubic  contents  of  the  drawers  of  a 
school  desk:  one  is  .965  meters  long,  .863  meters  wide,  and 
.127  meters  deep,  the  other  .91  meters  long,  .406  meters 
wide,  and  .127  meters  deep. 

6.  How  many  square  feet  in  the  surface  of  two  blocks  of 
stone,  one  4  ft.  in  each  dimension  and  the  other  3  ft.  long, 
2  ft.  4  in.  wide,  and  1  ft.  thick? 

7.  In  a  section  of  land  how  many  sq.  in.? 

7  Metric.  In  a  section  of  land  how  many  hektares? 

8.  In  a  pile  of  wood  16  ft.  long,  3-i  ft.  wide,  and  5  ft.  high, 
how  many  cd.? 

8  Metric.  In  a  pile  of  wood  4.87  meters  long,  1.06  meters 
wide,  and  1.52  meters  high,  how  many  steres? 

9.  If  a  field  260  rd.  long  contains  9|  A.  what  is  its 
width? 

10.  How  many  spoons  weighing  16  pwt.  11  gr.  can  be 
made  from  5  ft).  1  pwt.  11  gr.  of  silver? 

10  Metric.  How  many  spoons  weighing  25.649  grams 
can  be  made  from  1872.4  grams  of  silver? 

11.  From  7  yr.  take  1  mo.  2  wk.  3  da.  11  hr.  35  min.  42  sec. 

12.  At  12|  cents  apiece  what  cost  posts  to  fence  a  ranch 
480  rd.  long  and  330  rd.  wide,  posts  set  24|  ft.  apart? 

13.  What  will  it  cost  to  put  three  wires  around  the  same 
ranch,  if  the  wire  is  worth  5^  cents  per  ft),  and  weighs  1| 
ft),  to  the  rod  ? 

14.  A  bbl.  of  kerosene  holding  32  gal.  loses  Mi  by 
evaporation.  One  half  of  the  remainder  is  sold  at  $.29 
per  gal.,  ^  of  that  remainder  at  $.27  per  gal.,  and  8  gal.  3 
qt.  at  $.26  a  gal.,  and  the  balance  at  $.28  per  gal.  It  cost 
$.17  per  gal.     Find  the  gain. 

15.  Reduce  660  ft.  to  the  decimal  of  a  mile. 

16.  A  ranchman  buys  4  sets  of  harness  at  $31.75  apiece, 
a  wagon  for  $175,  a  bbl.  of  sugar  for  $17.50,  and  grain  sacks 
to  the  amount  of  $18.42;  how  much  wheat  at  $1.50  per 
cental  will  it  take  to  pay  the  bill  ? 


ARITHMETIC.  165 

17.  From  a  pile  of  wood  containing  8964  cu.  ft.,  9^  cd. 
were  sold  at  one  time  and  7^  at  another;  find  the  worth  of 
the  remainder  at  $7.25  i)er  cd. 

17  Metric.  A  pile  of  wood  contained  253.736413  steres; 
33.514492  steres  were  sold  at  one  time  and  27.173913  at 
another;  what  is  the  remainder  worth  at  $2,001  per  stere? 

18.  If  a  herder  averages  7  mi.  148  rd.  travel  in  a  day, 
how  much  does  he  travel  in  a  year? 

18  Metric.  If  a  herder  averages  1609.372  meters  a  day, 
how  far  does  he  go  in  a  year? 

19.  From  f  ft).+4f  oz.+31^  pwt,  take  (f  oz.— |  pwt.). 

20.  How  many  dollars  of  25.8  gr.  can  be  made  from  2 
ft).  6  oz.  17  pwt.  12  gr.  of  gold? 

20  Metric.  How  many  dollars  of  1.6753  grams  can  be 
made  from  961.5584  grams  of  gold? 

21.  If  6  cu.  yd.  2|  cu.  ft.  of  earth  are  used  in  grading 
one  rod  of  street,  how  much  will  be  used  in  grading  16| 
blocks,  allowing  12  blocks  to  a  mile? 

22.  A  milkman  starts  out  with  9  six-gallon  cans  of  milk. 
He  delivers  a  pt.  each  to  35  customers,  1  qt.  each  to  48,  2 
qt.  each  to  69.  He  sells  \  of  what  is  left,  lacking  one  pt., 
to  a  boarding-house  keeper;  how  much  remains  unsold? 

23.  How  many  cu.  ft.  in  a  wall  one  rod  long,  5^  ft.  high, 
and  1  ft.  thick? 

24.  How  many  lots  45  by  150  feet  can  be  made  from  10 
A.,  allowing  one  fourth  for  streets? 

25.  At  $1.60  an  ounce  what  is  the  value  of  2  doz.  spoons, 
each  weighing  11  pwt.  23  gr.  ? 

25  Metric.  At  $.50  for  31.168  grams  what  is  the  value  of 
2  doz.  spoons,  each  spoon  weighing  18.636  grams? 

26.  A  cistern  holds  98  bbl.  If  4  gal.  run  in  by  one  pipe 
in  a  minute,  and  6  gal.  run  out  in  the  same  time  by  another, 
how  long  will  it  be  in  emptying  ? 

27.  A  man  owning  a  quarter  section  of  land,  gave  a  piece 
17  rd.  square  as  a  church  site;  how  much  has  he  left? 


166  CALIFORNIA   SERIES. 

28.  What  must  be  paid  for  a  pile  of  wood  15  ft.  long,  4 
ft.  high,  4  ft.  wide,  at  $9.75  per  cd.  ? 

29.  The  small  wheel  of  a  bicycle  is  3  ft.  in  circumfer- 
ence, and  the  large  wheel  8  ft.  and  3  in.;  how  many  more 
times  does  the  small  wheel  turn  than  the  large  one  in  going 
a  mile? 

30.  From  -|  A.  take  79  sq.  rd.  7  sq.  yd.  6  sq.  ft.  98  sq.  in. 

31.  How  much  carpeting  |  of  a  yd.  wide  will  it  take  to 
carpet  a  room  14  ft.  by  27  ft.,  the  breadths  to  run  crosswise? 

32.  At  $.32  a  square  yard,  what  will  it  cost  to  plaster  a 
room  11  ft.  3  in.  by  15  ft.,  and  9  ft.  high,  deducting  one  half 
the  surface  of  two  doors  each  3  ft.  wide  and  6  ft.  8  in.  high, 
and  3  windows  each  2|-  ft.  wide  and  6  ft.  high  ? 

33.  How  many  cu.  ft.  in  a  tank  9  ft.  3  in.  long,  6  ft.  4  in. 
wide,  4  ft.  9  in.  deep,  inside  measurement? 

34.  From  ^  of  2  A.  159  sq.  rd.  13  sq.  yd.  4  sq.  ft.  138  sq. 
in.  take  100  sq.  rd.  24  sq.  yd.  7  sq.  ft.  96  sq.  in. 

35.  If  a  celebration  on  the  Fourth  of  July  begins  at  10 
o'clock  A.  M.  in  Chicago,  at  what  hour  must  it  begin  at  Los 
Angeles  to  be  at  the  same  time  ? 

36.  Buenos  Ayres  is  longitude  58°  22'  W.,  and  the  Cape 
of  Good  Hope  18°  28'  E. ;  when  it  is  6  hr.  30  min.  a.  m.  in 
Buenos  Ayres,  what  is  the  time  at  the  Cape  of  Good  Hope? 

37.  If  you  sleep  9  hr.  each  night,  what  decimal  part  of 
your  time  are  you  asleep? 

38.  Add  I  mi.,  i  rd.,  f  ft. 

38  Metric.  Add  1005.84  meters,  1.67  meters  and  .25 
meters. 

39.  If  a  tank,  containing  105  cu.  ft.,  is  7  ft.  long  and  4 
ft.  deep,  what  is  its  width? 

40.  At  6  cents  a  square  foot,  what  will  it  cost  for  the 
wainscoting  of  a  room  16  ft.  wide  and  22  ft.  long,  if  ih'e 
wainscot  is  2  ft.  10  in.  high,  deducting  for  3  doors,  which, 
with  their  casings,  are  each  4  ft.  11  in.  wide? 

41.  How  much  will  it  take  to  carpet  a  room  18  ft.  wide 


ARITHMETIC.  167 

■  22  ft.  long,  if  the  carpeting  is  |  yd.  wide,  and  the  breadths 
run  across  the  room? 

41  Metric.  How  much  wdll  it  take  to  carpet  a  room  5.48 
meters  wide,  6.70  meters  long,  with  carpet  .68  meters  wide? 

42.  How  much  carpet  a  yard  wide  will  carpet  a  room  11 
ft.  11  in.  wide,  and  17  ft.  10  in.  long,  if  the  breadths  run 
lengthwise  ? 

43.  How  much  will  it  cost  to  carpet  a  church  with  yard 
wide  carpeting  at  $1.50  per  yd.,  the  auditorium  being  60  ft. 
wide  and  80  ft.  long,  breadths  running  lengthwise,  and  10 
yd.  extra  allowed  for  the  pulpit  platform;  one  parlor  being 
20  ft.  wide  and  24  ft.  long,  the  other  20  ft.  wide  and  36  ft. 
long,  breadths  running  crosswise  in  both  parlors? 

44.  A  stage  is  robbed  of  two  bars  of  bullion  Aveighing  170 
lb.  each,  worth  $3700.00;  how  much  is  it  worth  an  ounce? 

45.  What  wdll  it  cost  to  carpet  a  room  17  ft.  wide  and  26 
ft.  long,  with  carpet  |  yd.  wide,  at  $2.75  per  yd.,  allowing 
yV  of  a  yd.  for  matching,  if  the  breadths  run  across  the 
room,  and  a  border  of  1  ft.  wide  is  used,  costing  $1.95  per 
yard?     How  much  border  will  it  take? 

46.  What  part  of  a  yard  is  yttt  of  a  mile? 

47.  Reduce  to  lower  denominations  |-  of  .225  of  a  mile. 

48.  Reduce  yf  of  a  cd.  ft.  to  the  fraction  of  a  cd. 

49.  What  will  it  cost  to  paper  a  room  16  ft.  wide,  22  ft. 
long,  9  ft.  high,  with  paper  at  $.87-|  per  roll,  8  yd.  in  a  roll 
and  H  ft.  wide,  allowing  20  sq.  yd.  for  doors,  windows,  and 
baseboards  ? 

50.  A  man  having  2  T.  7  cwt.  28  lb.  of  hay,  sold  5  cvd. 
91  ib.;  what  fraction  of  the  whole  did  he  sell? 

51.  A  man  has  a  piece  of  land  360  ft.  long  containing 
396  sq.  rd.  21  sq.  yd.;  he  is  offered  $605  an  acre;  but  he 
runs  a  10-foot  alley  lengthwise  through  the  piece  and  di- 
vides it  into  16  equal  lots,  which  he  sells  at  $175  each; 
what  is  the  size  of  his  lots,  and  what  does  he  gain? 


f^ 

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168  CALIFORNIA   SERIES, 


UNITED  STATES  MONEY. 

United  States  money  is  written  decimally,  the  dollar  be- 
ing the  unit.     Five  decimal  places  have  been  named. 

The  mill  is  not  represented  by  any 
coin.  Results  should  be  carried  to  hun- 
dredths only. 

Gouverneur  Morris  first  recommended 
making  our  money  in  decimal  divisions;  afterward  Jeffer- 
son and  Hamilton  improved  upon  his  plan. 

The  Spanish  silver  dollar  was  chosen  as  the  unit,  and 
coinage  commenced  in  1792. 

Gold  and  silver  are  soft,  hence  the  coins  are  now  alloyed 
with  y^Q-  of  some  other  metal  to  harden  them.  In  gold 
coins  the  alloy  is  a  mixture  of  silver  and  copper;  the  dollar 
weighs  25.8  grains,  and  all  other  coins  are  multiples  of  this 
weight.     They  are  2-|-,  3-,  5-,  10-,  and  20-dollar  pieces. 

Silver  coins  are  alloyed  with  copper.  The  dollar  weighs 
412^  grains;  the  smaller  coins  are  lighter;  thus,  2  half  dol- 
lars, 4  quarters,  10  dimes,  or  20  half  dimes  weigh  but  385.8 
grains.  The  half  dime  is  now  made  of  nickel  and  copper, 
and  the  2-  and  1-cent  pieces  of  copper. 

i=.m  i=.20  i=.SSi  |=.87i         j\=Mi 

i=Mi         i=AO  i=.14f         tV=-08^         TV-=-06i 

|=.66f         |=.60  i=.12i         fV=-41|         A-.18I 

3 T^i  1 1fi2  5 a<r>l  IX 012  _7_. 

4 -'-^  6" -L"3  8 ^-2  T2 '^^S  IG 


EXERCISE  199. 

1.  At  $.16f  per  pound,  what  will  96  ib.  of  coffee  cost? 

$.16i=$i.     96X'1^i-='n6. 

2.  How  many  doz.  eggs  at  $.12-^  per  doz.  can  be  bought 

for  $2.75? 

$.12^=$i     n.75~H=22  doz. 


ARITILVETIC.  169 

3.  What  will  128  centals  of  wheat  cost  at  $1.25  per 
cental? 

$.25=i     iXl28=32.     128+32=$160. 

4.  Find  the  cost  of  8576  bricks  at  $9  per  thousand. 

8576--1000=8.576 
8.576X9=$77.184 

5.  What  will  7864  ft.  of  hay  cost  at  $12.00  per  T.? 

7894--1000=7.894 
7.894--2=3.947  T. 
3.947  X12=$46.364. 

6.  AVhat  part  of  100  is  33^?     What  part  is  8|?     37^? 

7.  What  part  of  100  is  87^?     6f?     16;i? 

8.  What  is  i  of  100?    I  of  100?     |  of  100? 

9.  What  is  I  of  100?    yV  of  100?    j\  of  100? 

10.  What  part  of  100  is  411? 

11.  What  will  10  yd.  of  cotton  cloth  cost  at  12^  cents 
per  yd.? 

12.  Find  the  cost  of  6  centals  of  w^heat  at  $1.16f  per 
cental. 

13.  A  lady  paid  $12.00  for  cloth  at  $.16f  per  yard;  how 
many  yards  did  she  buy? 

14.  Find  the  cost  of  720  ft.  of  soap  at  $.08^. 

15.  How  many  yards  of  cloth  at  $1.33^  per  yard  can  be 
bought  for  $128.00? 

16.  What  will  248  dozen  eggs  cost  at  $.374  per  dozen? 

17.  At  $1.62|  apiece  how  many  histories  can  be  bought 
for  $2613.00? 

18.  What  will  7  cords  of  wood  cost  at  $9.75  per  cord  ? 

19.  At  $11  per  cwt.  how  much  sugar  can  be  bought  for 
$44? 

20.  What  will  500  bricks  cost  at  $8  per  thousand  ? 

21.  What  will  5500  ft.  of  boards  cost  at  $22  per  thousand  ? 

22.  At  $.06|  per  pound  how  much  honey  can  be  bought 
for  $16? 

23.  How  many  books  at  $1.25  apiece  will  $48.75  buy? 


170  CALIFORNIA   SEEIES. 

24.  What  will  874  grain  sacks  cost  at  6:j  cents  apiece? 

25.  When  beeswax  is  $.25  per  ib.  how  many  pounds  will 
$16  buy? 

26.  What  cost  84  lb.  of  veal  at  12-J  cents  per  pound? 

27.  At  $.12:|  per  yd.  what  will  648  yd.  of  gingham  cost? 

28.  How  much  alfalfa  seed  at  $.12^  per  pound  can  be 
bought  for  $19? 

29.  What  will  976  boxes  of  limes  cost  at  $.75  per  box? 

30.  What  Avill  be  the  cost  of  879  centals  of  wheat  at 
$1.40  per  cental? 

31.  At  $.87-2  pel*  yard  how  much  flannel  can  be  bought 
for  $40? 

32.  A  hotel  keeper  spends  $9.33^  for  spring  chickens  at 
$.66f  apiece;  how  many  does  he  buy? 

33.  What  will  376  dozen  eggs  cost  at  $.37-J  per  dozen? 

34.  At  6|  cents  apiece  how  many  paper  bags  can  a  con- 
fectioner buy  for  $32? 

35.  When  hay  is  $12.75  per  ton  how  many  tons  will  $357 
buy? 

36.  Find  the  cost  of  7212  lb.  of  evaporated  apples  at  8^ 
cents  per  ib. 

37.  At  $1.25  per  yard  what  will  18  yd.  of  silk  cost? 

38.  What  will  189  lb.  of  coffee  cost  at  $.33^  per  pound? 

39.  How  many  pounds  of  tea  at  $.62^  per  lb.  can  a  grocer 
buy  for  $26.87^? 

40.  How  many  sheep  at  $4.75  apiece  can  be  bought  for  $95? 

41.  At  $1.87^  per  sack  what  will  408  sacks  of  flour  cost? 

42.  At  $1,625  per  yd.  what  will  248  yd.  of  silk  cost? 

43.  A  dealer  buys  butter  for  $.87^  a  roll  to  the  amount 
of  $100;  how  many  rolls  does  he  buy? 

44.  What  will  249  yd.  of  velvet  cost  at  $2.66S  per  yard  ? 

45.  At  $.83^  per  yard  what  will  726  yards  of  cashmere 
cost? 

46.  What  will  97856  feet  of  boards  cost  at  $19.00  per 
thousand  ? 


ARITHMETIC.  171 

47.  Find  the  cost  of  785409  bricks  at  $8.00  per  thousand? 

48.  At  $6.00  per  cwt.  what  will  9856  ib.  of  beef  cost? 

49.  What  will  764398  lb.  of  flour  cost  at  $2.40  per  cwt.? 

50.  Find  the  cost  of  43986  lb.  of  coal  at  $.70  per  cwt. 

51.  A  man  paid  $95.00  for  freight  on  wool  at  $1.25  per 
cwt.;  how  much  wool  had  he? 

52.  What  is  the  freight  on  7543  lb.  of  merchandise  at 
$2.50  per  cwt.  ? 

53.  What  will  98756  lb.  of  hay  cost  at  $12.00  per  ton? 

54.  How  many  pounds  in  a  load  of  coal  which  costs  $8.00, 
when  coal  is  $12.00  per  T.? 

55.  What  is  the  value  of  3  loads  of  hay  weighing  respect- 
ively 1975,  1125,  and  1240  pounds,  at  $12.75  per  T.? 

56.  How  many  dollars  can  be  made  from  one  lb.  of  pure 
gold  ? 

57.  How  many  dollars  can  be  made  from  one  ib.  of  pure 
silver? 

58.  How  many  dollars  can  be  made  from  7  lb.  11  oz.  18 
pwt.  3  gr.  of  pure  silver? 

59.  How  many  dollars  can  be  made  from  1  lb.  1  pwt.  21 
gr.  of  pure  gold? 

60.  How  many  eagles  can  be  made  from  2  lb.  3  pwt.  18 
gr.  of  pure  gold? 

61.  How  many  half  dollars  can  be  made  from  6  lb.  6 
pwt.  18  gr.  of  pure  silver? 

62.  How  many  quarters  can  be  made  from  1  lb.  6  oz.  1 
pwt.  164  gr.  of  pure  silver? 

63.  How  many  dimes  can  be  made  from  3  lb.  3  pwt.  9 
gr.  of  pure  silver? 

64.  A  bar  of  silver  bullion  is  .975  pure;  how  many  half 
dollars  can  be  made  from  it?    Weight  9  lb.  2  oz.  8  pwt. 

65.  How  many  $2^  pieces  can  be  made  from  3  lb.  8  oz. 
11  pwt.  of  gold  bullion? 


172  CALIFORNIA   SERIES. 


GENERAL  ANALYSIS. 

When  several  successive  operations  in  multiplication  and 
division  are  to  be  performed  in  the  same  example,  each 
operation  may  be  determined  by  ordinary  analysis  and 
indicated  by  placing  the  number  concerned  above  or  below 
the  division  line.     Thus,  , 

If  15  acres  of  land  cost  $620,  what  are  12  acres  worth  at 
the  same  rate? 

OPERATION.  Analysis, — If  15  acres  cost  $620, 1  acre  costs 

-'-  ^  "^        '^  ^  of  $620,  or  ^^.     12  acres  cost  12  times  1 

$020x;2     ...r  ^^  ^^ 

-.rv-^=-^49  6  i|;620    ,^    $620X12       ^.  .     . 

ip  acre,  or -^r— xl2  =  -^ — r^ — .      The    work  is 

^  lo  lo 

shortened  by  cancellation. 

If  18  tons  of  coal  cost  $189,  how  many  tons  can  be  bought 
for  $105? 

OPERATION.  .  T^        1    J    n  i,  1 

Analysis. — For  1  dollar  you  can  buy  -^t^ 
2        5  -^  '  189 

18 
18X10^^         ^  of  18  tons,  or  — —  tons ;  and  for  105  dollars 

/V^  '  =^l  0  tons.  ^<^J 

189  T^-  ,.  18,  18x105. 

'^  ^  '  lOo  times  7—-,  tons,  or  — tt^—  tons. 

If  200  bushels  of  oats  will  last  30  horses  50  days,  how 

long  will  150  bushels  last  45  horses? 

operation.  Analysis. — 200  bushels  last   50 

2  5       JL^  days;   1  bushel  will  last  ^  of  50 

^0X;f50X'S0     ^^    ,  50 

200X45 ~        days,  days,  or  ^-^^  days,  and  150  bushels, 

o  Q  ,_^  ,.  ,  50x150 

f  p  loO  tunes  as  many  days,  or  — ^^^ — 

days ;  this  is  the  number  of  days  for  30  horses,  for  1  horse  30  times 

,  50x150x30  ,  A  t      Ar  X.  1 

as  many  davs,  or .^rr; days,  and  for  45  horses  -r=  as  many 

J       .  200  '  45  "^ 

,  ,  50x150x30  , 

days  as  one  horse,  or  — -^-^r^ — r^r—  davs. 
•^  200  x4o 


ARITHMETIC.  173 

Observe  that  in  these  examples,  one  number  is  of  the 
same  kind  as  the  answer  sought,  and  the  others  are  in 
hke  pairs.  In  the  last  example,  50  is  like  the  answer,  days. 
Then  there  are  two  numbers  of  bushels,  and  two  of  horses. 
The  first  two  examples  have  one  pair  each,  beside  the  num- 
ber tliat  is  like  the  answer. 

The  following  is  a  short  statement  of  the  method:  Begin 
with  the  number  like  the  answer;  then  in  each  pair  reason 
from  the  given  number  of  the  pair  to  1,  and  from  1  to  the 
number  required. 

The  results  all  the  way  through  are  like  the  required 
answer. 

EXERCISE  200.    (Written.) 

1.  If  18  sheep  are  worth  $45,  what  are  30  sheep  worth  at 
the  same  rate  ? 

2.  If  7  men  dig  a  ditch  in  15  days,  how  long  will  it  take 
15  men? 

3.  If  48  rods  of  fence  cost  $108,  what  wall  84  rods  cost? 

4.  If  a  locomotive  goes  564  miles  in  24  hours,  how  far 
will  it  go  in  22  hours? 

5.  If  160  A.  of  land  produce  96  tons  of  wheat,  how  many 
tons  will  175  A.  produce? 

6.*  If  75  A.  of  land  produce  50  tons  of  wheat,  how  many 
A.  will  produce  18  tons?  .^ 

7.  If  50  chairs  cost  $112.50,  how  many  chairs  can  be 
bought  for  $90? 

8.  If  8f  yd.  of  cloth  cost  $17.50,  what  will  12^  yd.  cost? 

9.  If  12  men  earn  $78  in  4  days,  how  many  men  will  earn 
$58-|  in  the  same  time? 

10.  If  18  men  can  do  a  piece  of  work  in  32  days,  how 
many  men  will  do  it  in  24  days? 

11.  If  the  freight  for  transporting  18  cwt.  of  household 
goods  from  San  Jose  to  Los  Angeles  is  $61.20,  what  will  it 
cost  to  transport  42  cwt.  ? 

12.  If  30  gal.  of  oil  cost  $3.75,  what  cost  100  gal.? 


174  CALIFORNIA    SERIES. 

13.  If  a  man  can  perform  a  journey  in  14  days  of  10 
hours  each,  how  many  days  of  12  hours  each  will  he  need 
to  do  the  same? 

14.  If  12  cows  can  be  bought  for  $486,  for  how  much  can 
22  cows  be  bought? 

15.  When  12  cows  cost  $486,  how  many  cows  can  be 
bought  for  $891? 

16.  If  9^  yd.  of  broadcloth  cost  $44^,  how  many  yd.  will 
cost  $33i  ? 

17.  If  it  costs  $720  to  transport  12  tons  of  freight  480 
miles,  what  will  it  cost  to  transport  15  tons  300  miles? 

18.  If  16  men  earn  $640  in  4  wk.,  Avhat  will  18  men  earn 
in  2  wk.? 

19.  If  130  ft),  of  tea  cost  $117,  what  will  80  ft),  cost? 

20.  If  a  pasture  will  feed  120  horses  81  da.,  how  many 
horses  will  it  feed  108  da.? 

21.  If  12  men  in  12  da.  of  9  hr.  each  can  perform  a  cer- 
tain piece  of  work,  how  many  days  of  8  hr.  each  will  it  take 
9  men? 

22.  How  many  lb.  of  sugar  can  you  buy  for  $380  if  20 
ft),  cost  $1.90? 

23.  If  it  takes  27  yd.  of  carpeting  f  of  a  yd.  wide  to  car- 
pet a  certain  room,  how  many  yards  of  1  yd.  wide  carpeting 
will  it  take  ?  » 

24.  If  9i\  yd.  of  cloth  -|  of  a  yd.  wide  cost  $11.40,  what 
will  10  yd.  1^  yd.  wide  cost? 

25.  If  6  bbl.  of  flour  last  80  men  12  da.,  how  long  will  9 
bbl.  last  60  men? 

26.  If  a  doz.  brooms  cost  $4.50,  how  many  brooms  can 
you  get  for  $3.37^? 

27.  IIow  many  men  will  build  35  rd.  of  wall  in  the  same 
time  that  6  men  build  42  rd.? 

28.  If  7  men  dig  a  ditch  28  feet  long  in  2  da.  of  8  hr. 
each,  how  many  da.  of  10  hr.  each  will  it  take  10  men  to 
dig  a  ditch  25  ft.  long? 


ARITHMETIC.  175 

29.  If  it  cost  $42  to  plaster  the  ceiling  of  a  room  14  ft. 
long  and  12  ft.  wide,  what  will  it  cost  for  a  room  16  ft.  long 
and  14  ft.  wide? 

30.  If  11^  lb.  of  coffee  cost  $3.45,  what  will  10^  lb.  cost? 

31.  When  the  shadow  of  a  post  10  ft.  6  in.  high  is  12  ft.  3 
in.  long,  what  is  the  length  of  shadow  of  a  post  8  ft.  9  in. 
high  ? 

32.  The  shadow  of  a  post  16  ft.  3  in.  high  is  5  ft.  5  in. 
long;  what  height  of  post  will  give  a  shadow  3  ft.  4  in. 
long? 

33.  If  4  men  huild  12^  rd.  of  fence  in  3^  da.,  how  long 
will  it  take  18  men  to  build  237i%  rd.? 

34.  If  a  tank  36  in.  long,  22  in.  wide,  and  7  in.  deep 
holds  24  gal.,  how  much  will  a  tank  3  ft.  8  in.  long,  1  ft.  2 
in.  wide,  and  1  ft.  deep  hold  ? 

35.  If  H  of  an  acre  of  land  is  worth  $198,  what  are  -J  of 
an  acre  worth? 

36.  At  the  rate  of  14  lb.  for  $1,  what  will  8  bbl.  of  sugar 
averaging  259  lb.  to  a  barrel  cost? 

37.  How  many  oranges  at  15  cents  a  dozen  will  pay  for 
7  5-gallon  cans  of  kerosene  at  $f  a  gallon  ? 

38.  I  buy  a  certain  quantity  of  rice  at  $4.50  per  lOO  lb. 
and  pay  for  it  with  717  ft.  of  pine  lumber  at  $15  per  M; 
what  weight  of  rice  did  I  buy? 

39.  A  farmer  bought  grain  bags  worth  7-i  cents  each  for 
150  sacks  of  oats  averaging  125  lb.  each  at  $1.20  a  cental; 
how  many  grain  bags  did  he  receive  ? 

40.  Sold  a  newspaper  proprietor  3  bun.  of  paper,  60  lb. 
each,  at  7  cents  per  pound,  for  which  he  agreed  to  furnish 
me  his  daily  paper  delivered  at  15  cents  per  week;  how 
long  did  I  receive  his  paper  ? 

41.  Bought  12  doz.  glass  jars  at  $1.75  and  paid  for  them 
in  potatoes  at  1^  cents  a  lb. ;  how  many  80-pound  sacks  did 
I  give  ? 


176  CALIFORNIA   SERIES. 


PROPORTION. 

Examples  in  General  Analysis  will  be  seen  to  contain  one 
number  of  the  same  kind  as  the  thing  required  in  the 
answer,  while  the  other  numbers  are  arranged  in  pairs. 

A  formula  or  statement  called  a  Proportion  is  sometimes 
used  in  such  examples,  to  precede  the  performing  of  the 
work  and  take  the  place  of  the  logical  and  proper  analysis 
of  the  example. 

3  days  is  what  fraction  of  12  days?  $4  is  what  fraction 
of  $24  ?     9  horses  of  16  horses  ? 

A  ratio  is  a  fraction  whose  terms  are  of  the  same  kind. 
Thus,  J 2"  o^'  4  expresses  the  ratio  of  3  to  12,  or  of  3  days  to 
12  days. 

Review  Exercise  137,  examples  1  to  16,  reading,  "  What 
is  the  ratio  of  8  to  20?"  Substitute  concrete  terms;  thus, 
8  men  to  20  men. 

A  ratio  is  often  written  by  using  two  dots  between  the 
terms.     Thus,  the  ratio  of  8  to  20  is  written  8 :  20. 

What  is  the  ratio  of  6  to  9  in  its  lowest  terms?  Of  12  to 
18?  What  can  you  say  of  these  two  ratios?  We  will  write 
them  equal.     f-:||,  or  (6  :  9)  =  (12  :  18). 

Two  equal  ratios  form  a  proportion. 

In  the  written  expression  four  dots  ( :  : )  are  often  used  for 
the  sign  =. 

The  first  and  last  terms  of  a  proportion  are  the  extremes ; 
the  second  and  third,  the  means. 

In  any  proportion  the  product  of  the  means  equals  the 
product  of  the  extremes;  thus,  9X1^=6X18.  Hence,  the 
product  of  the  means  divided  by  one  extreme  will  give  the 
other  extreme;  or,  the  product  of  the  extremes  divided  by 
one  mean  will  give  the  other  mean.     Thus, 


ARITHMETIC.  Ill 

9X12_  9X12     ^       6X18__  ^         6X18 _ 

'    6     ~     '  18    ~''  9     ~^^-  12 


To  determine  and  write  a  proportion. 

If  15  A.  of  land  are  worth  $620,  what  are  12  A.  worth? 

OPERATION.  Explanation. — $620    is    like     the    required 

15:12::620:?    answer.     15  and  12  are  A.     The  ratio  of  15  A. 
12x620  to  12  A.  must  be  the  same  as  the  cost  of  15  A., 

=4  J  6.     ^(320,  to  the  cost  of  12  A.;  or,  as  written  above. 
Soh^e  as  in  Anah^sis. 


15 


To  arrange  the  terms, 

1.  Place  the  number  which  is  hke  the  required  answer 
for  the  third  term; 

2.  If,  in  the  nature  of  the  problem,  the  answer  ought  to 
be  larger  than  the  third  term,  arrange  the  pair  so  that  the 
second  term  shall  be  larger  than  the  first,-  but  if  the  answer 
should  be  smaller  than  the  third  term,  let  the  smaller  of 
the  two  numbers  be  the  second  term.  Then  divide  the 
product  of  the  means  by  the  given  extreme. 

Sometimes  the  result  depends  upon  the  relations  of  sev- 
eral pairs,  producing  a  compound  proportion.  In  such  case 
consider  the  result  with  reference  to  each  pair  separately, 
as  in  simple  proportion. 

Thus,  in  the  third  analysis.  General  Analysis,  consider 
first  the  horses  alone;  then  the  bushels  alone. 

45:  30    1  _-^  . 
200: 150  \  — ^^-  • 

30X150X50_^K 
45X200         •     ' 

For  work,  perform  the  examples  in  General  Analysis. 
12— A 


178  CALIFORNtA   SERIES. 


PARTNERSHIP. 

Two  men  paid  $6  for  a  pasture  1  month.  If  each  puts  in 
2  coAVS  what  should  each  pay?  If  one  puts  in  2  cows  and 
the  other  1  cow  what  should  each  pay?  If  the  first  had 
his  2  cows  in  pasture  4  weeks  and  the  second  his  1  cow 
only  2  weeks  how  much  should  each  pay? 

Thus,  we  see  that  each  man's  share  of  the  expense  de- 
pends upon  the  product  of  the  number  of  cows  pastured 
and  the  time,  if  the  times  are  different. 

An  association  of  two  or  more  persons  together  in  busi- 
ness is  called  Partnership. 

The  persons  associated  are  called  partners. 

The  money  subscribed  is  called  the  capital  or  stock. 

Each  partner  receives  the  same  part  or  fraction  of  the 
losses  or  gains  that  his  capital  is  of  the  whole  capital  in- 
vested, if  all  invest  for  the  same  time;  if  for  different  times, 
each  partner's  capital  must  be  nuiltiplied  by  the  time  it  is 
in  use  and  the  product  taken  as  his  share  of  the  capital; 
the  sum  of  these  products  being  taken  as  the  entire  capital, 
provided  no  special  division  has  been  agreed  upon. 

Property  of  all  kinds  owned  by  a  firm  are  its  assets. 

Its  debts  are  liabilities. 

EXERCISE  201.   (Written.) 

Find  out,  if  you  can,  how  to  prove  these  examples  and 
prove  each. 

1.  Two  men  enter  into  partnership  in  the  grocery  busi- 
ness. A  furnishes  $2500  capital;  B,  $1500.  Their  gain 
the  first  year  was  $1840.     Find  the  share  of  each. 

2.  A  and  B  trade  together.  A  furnishes  -^  of  the  capi- 
tal; B,  the  remainder.     Divide  their  loss  of  $637  fairly. 

3.  Two  men  hire  a  pasture  for  $96.     One  pastures  40 


ARITHMETIC.  179 

sheep  for  11  weeks;  the  other,  Qi)  sheep  8  weeks.     What 
should  each  pay? 

4.  Two  men  engaged  in  the  clothing  business  with  a  joint 
capital  of  '$6000.  The  first  year's  gain  was  $2892,  of  which 
one  received  $964.  AVhat  amount  of  capital  did  each  furnish? 

5.  Three  men  engage  in  business.  A  puts  in  $2000  the 
first  of  January;  B  $3000  the  first  of  March;  and  C  $4000 
the  first  of  April.  The  profits  at  the  close  of  the  year  of 
$6045  will  be  shared  how  ? 

6.  Divide  $195  among  3  boys,  giving  them  3,  4,  and  6 
parts  respectively. 

7.  A  bankrupt  owes  A  $1000,  B  $1500,  C  $1800,  D  $2000, 
and  E  $2700.  His  assets  are  $6000.  What  sum  can  he 
pay  each? 

8.  A  man  has  $5175  and  owes  $6210;  what  can  he  pay 
on  every  $1  he  owes?  What  will  a  man  to  whom  he  owes 
$1320  receive? 

9.  A,  B,  and  C  sent  a  ship  loaded  with  Wellington  coal 
to  San  Francisco.  A  put  on  board  180  tons,  B  250  tons, 
and  C  400  tons.  On  account  of  storm  249  tons  Avere  thrown 
overboard;  find  the  loss  of  each. 

10.  In  a  certain  firm  B  has  3  times  as  much  capital  as 
A,  and  C  has  ^  as  much  as  the  other  two.  What  is  each 
one's  share  in  a  loss  of  $786? 

11.  In  a  gain  of  $600  A  received  -i,  B  |-,  and  C  the  re- 
mainder. If  the  whole  capital  was  12  times  A's  gain  what 
was  the  capital  of  each? 

12.  Two  men  receive  $1000  for  grading.  One  furnishes 
3  teams  20  days  and  the  other  5  teams  30  days.  If  the 
first  receives  $100  for  overseeing  the  work  what  does  each 
receive  ? 

13.  Two  men  contract  to  move  5316  cu.  yds.  of  gravel  at 
25  cents  a  cu.  yd.,  and  agree  to  share  the  profits  in  the  pro- 
portion of  2  to  3.  They  employ  5  teams  45  days  at  $4  each 
per  day.     What  did  each  make  ? 


180  CALIFORNIA    SERIES. 

14.  Divide  1728  in  the  proportion  of  3,  4,  and  5. 

15.  Three  men  have  wheat  of  different  grades  in  a  ware- 
house. A  has  1200  centals  worth  $1.10;  B  has  800  centals 
worth  $1.25;  C  has  1600  centals  worth  $1.12^  The  wheat 
being  damaged,  the  whole  was  sold  for  $3090.  Find  each 
one's  share. 

EXERCISE  202.    (Oral.) 

1.  Divide  75  cents  among  3  boys,  giving  to  the  first  3 
cents  as  often  as  to  the  second  5  cents  and  the  third  7  cents. 

2.  Albert  and  James  buy  a  book  together  costing  $1.50, 
of  which  Albert  paid  50  cents  and  James  the  rest.  They 
afterwards  sell  it  for  75  cents.     What  should  each  receive? 

3.  A  lady  gave  $2  to  her  children  aged  8  and  12  in  pro- 
portion to  their  ages.     What  did  each  receive? 

4.  Three  girls  bought  $l's  worth  of  oranges;  the  first 
receiving  ^,  the  second  ^,  and  the  third  the  rest.  How 
much  money  did  each  contribute? 

5.  I  hire  a  pasture,  in  company  with  a  friend,  for  $65. 
I  pasture  8  cows  4  months;  my  friend,  11  cows  3  months. 
What  is  his  share  of  the  expense? 

6.  Divide  50  cents  in  the  proportion  of  \  and  -|. 

7.  I  owe  $2000  and  have  $1500.  How  much  can  I  pay 
for  every  dollar  owed  ? 

8.  What  will  a  man  whom  I  owe  $100  receive? 

9.  A  man  leaves  $5000  to  his  two  sons  in  the  inverse 
ratio  of  their  ages,  15  and  10.     Find  what  each  had. 

10.  Two  men  in  partnership  lose  $800,  of  which  the  first 
bears  $500.  Their  capital  was  $2400;  what  capital  did 
each  furnish? 

11.  Divide  45  marbles  with  two  companions  so  that  one 
shall  receive  2  to  your  1,  and  the  other  3  to  your  2. 

EXERCISE  203.    (Written.) 

Form  10  Partnership  examples  of  your  own,  perform, 
and  bring  to  the  class  for  dictation. 


ARITHMETIC.  181 


PERCENTAGE. 

Review  Exercises  139,  137,  123,  and  examples  13  to  20, 
Exercise  136. 

W  nat  is  y  o^-Q  01  Duu  .'^     ttju-     tfo-     too-     loo-      loo- 
lo  is  y-Q^  or  wiiai :     yoo-      lOO-      loo-      loo-      loo- 

Per  cent  means  hundredths  {irom.'per  centum,  by  the 
hundred).  Thus,  yf  o,  or  .03,  is  3  per  cent.  y^o>  oi'  -O'^^  is 
5  per  cent. 

The  word  rate  is  sometimes  used  for  per  cent. 

That  number  of  which  anothei*  is  a  fraction  or  per  cent 
is  called  the  base. 

Name  the  base  in  the  above  illustrations. 

Rewrite  the  above  illustrations,  using  the  term  per  cent 
instead  of  the  denominator  100,  with  the  answers  following. 

Thus,What  is  1  per  cent  of  600 f    Ans.  6. 

The  sign  %  means  per  cent. 

EXERCISE  204.    (Written.) 
Rewrite  the  following  in  decimal  and  common  forms, 
and  reduce  the  common  form  to  lowest  terms. 
Thus,  Q%=.OQ=jU--^i\. 


6  per 

cent. 

14f% 

5|% 

Qi% 

i% 

9    " 

4S% 

1\% 

^i% 

i% 

12    " 

114% 

^% 

m% 

2\% 

15    '' 

n% 

^% 

16|% 

%% 

18    '' 

i% 

5|% 

30% 

\% 

22    '' 

\\% 

22h% 

SSi% 

^% 

27    " 

12% 

17i% 

37^% 

Si% 

32    " 

85% 

13^% 

m% 

i% 

36    " 

84% 

23i% 

66f% 

87i% 

Q  1      " 
^TT 

96% 

21i% 

83i% 

i% 

Practice  on  the  last  two  columns  until  familiar. 


182  CALIFORNIA   SERIES. 

EXERCISE  205.    (Oral.) 
Name  the  corresponding  fractions  in  lowest  terms. 


b%           25% 

45% 

65% 

85% 

4% 

7%           30% 

50% 

70% 

90% 

100% 

10%            35% 

55% 

75% 

95% 

i% 

20%            40% 

60% 

80% 

2% 

t\% 

EXERCISE 

206. 

(Written.) 

How  many  lOOths,  or  %, 

are 

i       i 

5  3 

6  800 

2 

7 

7 
1  1 

1 

16  0 

3 
41 

2  _1_ 

3  1  2 

1                       3 

16                  400 

5 

T9- 

4 
9 

1  3 

2  00 

7 
45 

"1          ^io" 

5                        7 
8                     400 

4 

T3" 

5 
16 

1  1 

900 

12 
61 

"e                      8                  300                 150 

3 

22 

1  1 
12 

4 
21 

9 
¥0- 

Drill  on  the  first  four  columns  until  familiar. 

EXERCISE  207.    (Oral.) 
How  many  lOOths,  or  %,  are: 


i 

1 
25" 

1 

80 

7 
50 

4 
5 

3 
40 

f 

7 
"21T 

i 

1 
30 

3 
4 

3 
5 

9 
TO" 

4J 

5 

24 
"25" 

1 

5" 

1 
4-0" 

2 
"5 

tV 

7 
25 

4 

25 

6 
25 

1  1 
50 

1 
TO" 

^V 

A 

1  1 

2  0 

13 
20 

21 
50 

li 

^ 

A 

tV 

3 

2  0 

8 

"25" 

9 
5  0 

19 
2  0 

li 

4  0 

EXERCISE   208.    (Written.) 

Change  the  fractions  in  Exercise  123  to  lOOths,  and  re- 
write the  examples,  using  ''per  cent"  instead  of  "  lOOths." 
Thus,  in  Example  1,  |=tW-  hence,  rewrite  thus, 

What  cost  75  per  cent  of  a  yard  of  cloth  at  20  cents  a  yard? 

Same  with  Exercise  160,  orally. 

EXERCISE  209.    (Written.) 

Rewrite  in  fractional  form,  and  analyze,  using  the  deci- 
mal and  common  forms: 

1.  $75  is  3  per  cent  of  what  sum? 

2.  What  is  25%  of  $1728? 

3.  $750  is  20%  of  what? 

4.  $640  is  what  per  cent  of  $3200? 


ARITHMETIC.  183 

5.  What  is  t%  of  $9900? 

6.  $75  is  1^%  of  what  sum? 

7.  $25.92  is  U%  of  what  sum? 

8.  $102.50  is  what  %  of  $20500? 

9.  What  is  62^  per  cent  of  $7288? 

10.  What  is  16f%of  $36? 

11.  $490  is  what  %  of  $5000? 

12.  $6.50  is  12^%  of  what  sum? 

13.  What  is  40  per  cent  of  $1683.25? 

14.  $150  is  33^%  of  what? 

15.  $729.80  is  66|  per  cent  of  how  much? 

16.  $2.50  is  what  per  cent  of  $20? 

17.  What  is  2^X  of  $400? 

18.  What  is  14%  of  $1500? 

19.  $13.50  is  what  per  cent  of  $81? 

20.  $37.50  is  6%  of  what  amount? 

EXERCISE  210.    (Oral.) 
Perform  Exercise  137,  changing  each  answer  to  lOOths, 

or  %. 

EXERCISE  21  1.    (Written.) 

Change  the  fractions  in  Exercise  138  to  lOOths,  and  re- 
write the  examples,  using  per  cent  instead  of  lOOths.   ^ 


PRACTICAL  WORK  IN  PERCENTAGE. 

^  .  ,  c  J.— What  is  I  (.75)  of  16  ? 

General  forms  \  , 

r     r)  i         -^  K. — 12  is  4  (.75)  of  what  number? 

lor  rercentage:  ; 

I  L.— 12  is  what  fraction  (%)  of  16  ? 

(Compare  with  *'  General  Forms,"  p.  93.) 
EXERCISE  212.    (Written.) 

Change  the  per  cents  to  fractions  in  their  loAvest  terms, 
rewrite  in  general  form ,  and  analyze: 

1.  A  man  having  $3300  lost  3  per  cent  of  it;  how  much 
did  he  lose? 


184  CALIFORNIA   SERIES. 

2.  A  man  had  an  annual  income  of  $2500.  He  spent  10 
%  of  it  for  board;  5%  for  clothing,  and  18%  for  inciden- 
tals; how  much  did  he  spend  for  each? 

3.  A  man  lost  $120,  which  was  40%  of  all  he  had;  how 
much  had  he? 

4.  A  man  having  $5800  worth  of  hay  lost  $870  worth  by 
fire;  what  fraction  and  what  per  cent  of  the  whole  was  the 
part  lost? 

5.  If  you  buy  eggs  at  20  cents  a  dozen  and  sell  them  at 
a  gain  of  2|  cents  a  dozen,  what  fraction  and  what  per  cent, 
of  the  cost  do  you  gain  ? 

6.  A  merchant  sells  a  barrel  of  flour  for  $6.25,  which  was 
125%  of  what  it  cost  him;  what  did  it  cost  him? 

7.  A  jeweler  sold  a  watch  for  $36,  which  was  90  per  cent 
of  its  cost;  find  the  cost. 

8.  A  ship  carrying  8750  tons  of  coal  sprung  a  leak,  on 
account  of  which  it  was  found  necessary  to  throw  over- 
board 1250  tons;  what  per  cent  of  the  coal  was  thus  lost? 

9.  A  man  spent  in  one  year  $2150,  which  was  5f%  of 
what  he  had;  how  much  had  he? 

^10.  My  salary   is  $2400;    if  I    spend  S7i%   of  it,  how 
much  money  do  I  spend? 

EXERCISE  213.   (Oral  Analysis.) 

1.  A  man  having  800  boxes  of  oranges  lost  S%  by  decay; 
how  many  boxes  did  he  lose  ? 

2.  $25  is  25%  of  what  sum? 

3.  In  a  school  of  150  pupils  3  were  absent;  what  per  cent 
was  absent? 

4.  A  man  having  spent  33^%  of  his  money  has  $600 
left;  what  had  he  at  first? 

5.  A  boy  increasing  his  money  by  25%  of  itself  has  $1; 
what  had  he  at  first? 

6.  A  man  owning  75%  of  a  ship  sold  33^%  of  his  share 
for  $6000;  find  the  value  of  the  ship. 

7.  20  is  40%  of  what  number? 


ARITHMETIC.  185 

8.  Sold  a  horse  for  •I'lOO  at  20%  above  cost;  find  the  cost. 

9.  $18  is  what  per  cent  of  $72? 

10.  ^Bought  a  cow  for  $35  and  sold  her  for  20%  above 
cost;  what  did  1  receive  for  her? 


PEOFIT  AND  LOSS. 

Gains^  losses,  and  selling-price,  are  always  a  per  cent  or 
fraction  of  the  cost.  , 

The  cost,  then,  in  Profit  and  Loss,  is  always  the  base. 

EXERCISE  214.    (Written.) 

Label  everything  given  in  the  first  10  examples,  with  the 
word  gain,  loss,  selling-price,  or  cost,  rewrite  in  general  form, 
and  then  perform. 

1.  A  man  sold  a  harness  for  $35,  gaining  4Q%  on  the 
cost;  find  the  cost. 

Model:     $35  =  S.  P.     100%  =  Cost.     40%  =  Gain. 
$33  is  140%  of  wliaf  niirtiber? 

2.  I  wish  to  make  Zl\?4  on  a  ton  of  hay  which  cost  me 
$7.20;  for  what  must  I  sell  it? 

3.  By  selling  a  house  for  $3500  I  lose  $500  on  the  cost; 
what  fraction,  and  per  cent,  of  the  cost  did  I  lose? 

4.  A  merchant  sells  cloth  for  $3.75,  losing  16|/^;  what 
was  the  cost? 

5.  A  broker  bought  cotton  to  the  amount  of  $3840.  The 
price  falhng,  he  was  obliged  to  sell  at  2h%  loss;  find  his 
loss  and  selling  price. 

6.  A  man  bought  144  pounds  of  sugar  at  the  rate  of  12 
pounds  for  a  dollar  and  sold  it  at  10  cents  a  pound.  What 
per  cent  did  he  gain  ? 

7.  Bought  tea  at  37^  cents  and  sold  it  at  50  cents.  What 
was  gained  per  cent? 

8.  Sold  wheat  at  $1.05,  losing  \2\%;  what  did  it  cost? 


186  CALIFORNIA    SERIES. 

9.  A  merchant  marked  cloth  at  25%  advance  on  the 
cost.  The  goods  being  damaged,  he  was  obHged  to  take 
off  20%  of  the  marked  price,  selhng  it  at  $1.50  per  yard; 
what  was  the  cost? 

10.  I  sold  I  of  an  acre  of  land  for  what  the  whole  acre 
cost  me ;  what  was  my  gain  %  ? 

rl.  What  per  cent  is  gained  in  buying  goods  by  long 
ton  weight  and  selling  them  at  the  same  price  per  ton  by 
short  ton  weight  ? 

:    "ii^.  If  20%  is  lost  by  selling  wheat  at  $1,  for  what  must 
it  be  sold  to  gain  10.%  ? 

13.  By  selling  a  cow  for  $7  less  than  she  cost  I  lose 
I4y% ;  what  was  her  cost  and  selling  price  ? 

14.  How  shall  a  merchant  mark  cloth  that  cost  16| 
cents  per  yard  so  as  to  gain  20%  ? 

15.  I  buy  a  box  of  oranges  containing  300  oranges  for  $1.50; 
for  how  much  must  I  sell  them  per  dozen  to  gain  41|%'? 

16.  Sold  goods  for  $3.50  less  than  cost  and  lost  14%; 
what  should  I  have  gained  per  cent  by  selling  for  $2.75 
above  cost? 

17.  A  man  sold  a  sack  of  potatoes  at  a  loss  of  12^%, 
thereby  losing  10  cents;  find  the  cost. 

18.  A  man  sold  a  buggy  for  ll-|-%  above  cost,  and  with 
the  money  bought  another  which  he  sold  for  $160,  losing 
1H%.  Did  he  gain  or  lose  on  the  whole  and  how  much 
per  cent? 

19.  Bought  12  acres  of  land  for  $840.  Sold  i  of  it  at 
$85  per  acre,  i  of  it  at  $75  per  acre,  and  the  remainder  at  a 
loss  of  14f  %  on  an  acre;  what  per  cent  was  gained  or  lost 
on  the  whole? 

20.  Two  sets  of  furniture  were  sold  at  $35  each.  On  one 
there  was  a  gain  of  16|%;  on  the  other  a  loss  of  16|%; 
was  there  a  gain  or  loss  on  both,  and  how  much  %? 

"S^^  A  merchant  bought  carpetings  at  75  cents,  95  cents, 
and  $1.10;  for  what  must  he  sell  each  to  make  20%? 


ARITHMETIC.  187 

^^.  A  furniture  dealer  sold  10  dozen  chairs  for  $96;  if  he 
paid  55  cents  apiece  for  them,  and  5  cents  each  for  trans- 
portation, what  %  was  his  profit? 

^^An  oil  company  paid  8  cents  a  gallon  for  a  cask  of 
crude  oil  containing  31-|  gallons;  if  11^%  of  it  leaked  out, 
at  what  price  must  it  be  sold  per  gallon  to  gain  11^%  on 
the  cost? 

24.  A  grocer  sells  -f  of  a  barrel  of  sugar  for  $7.82,  losing 

8% ;  for  how  much  must  he  sell  the  remainder  to  gain  8% 

on  the  whole  ? 

Y^  xlk.  By  selling  a  suit  of  clothes  for  b%  less  than  cost,  a 

^  I    tailor  gets  $5.55  l?ss  than  if  he  had  sold  them  for  10% 

\    above  cost;  find  the  cost. 
*j       2^.  The  labor  in  making  a  machine  will  cost  $37.50,  and 
*^jthe  whole  cost  is  $65;   the  laborers  strike  and  get  an  ad- 
vance of  105'o  on  their  wages;  for  what  must  the  machine 
be  sold  to  gain  20%  ? 

2i.  A  merchant  bought  wheat  at  96  cents  a  cental,  and 

marked  it  for  sale  at  $1.12^.     He  afterwards  marked  up 

^the  price  6|%,  and  sold  240  centals.     The  buyer  failed, 

lowever,  and  settled  by  paying  75  cents  for  every  dollar  he 

owed.     Did  the  merchant  gain  or  lose,  how  much,  and  how 

much  per  cent? 

^.  A  hardware  merchant  bought  three  dozen  agate  ba- 
sins at  the  rate  of  3  for  $5,  and  sold  them  at  a  gain  of  $10 
on  the  whole;  what  was  the  average  selling  price  of  each, 
and  what  was  the  gain  per  cent? 

^9.  A  merchant  sold  25  yards  of  cloth  for  $31.25,  at  a 
loss  of  161%";  find  the  cost  per  yard. 

'  30.  A  boy  bought  oranges  at  40  cents  a  hundred,  lost  b% 
by  decay,  and  sold  them  at  the  rate  of  3  for  2  cents;  what 
was  his  gain  %  ? 

EXERCISE  215.    (Oral.) 

1.  A  man  bought  a  horse  for  $75  and  sold  him  at  a  gain 
of  20%;  find  the  selling  price. 


188  CALIFORNIA   SERIES. 

2.  Find  the  gain  per  cent  on  sugar  bought  at  8  cents  and 
sold  at  9  cents. 

3.  Sold  calico  at  16  cents,  gaining  4  cents;  what  was  the 
gain  %? 

4.  I  wish  to  make  ol\%  on  a  suit  of  clothes  that  cost 
$16;  for  what  must  I  sell  them? 

5.  A  grocer  sold  tea  for  30  cents  a  pound;  if  he  lost  16f  X, 
what  did  the  tea  cost? 

6.  Sold  a  carriage  for  $40  less  than  cost,  losing  40% ;  find 
the  cost. 

7.  If  a  dozen  lemons  are  bought  for  25  cents  and  sold  for 
35  cents,  what  is  the  per  cent  of  gain? 

8.  Gained  10  cents  by  selling  a  penknife  at  25%  profit; 
what  did  it  cost? 

9.  Sold  goods  for  \  more  than  I  paid  for  them;  what  was 
the  gain  %  ? 

10.  A  grocer  makes  10%  by  selling  coffee  at  2|  cents 
above  cost;  what  is  the  cost  and  the  selling  price? 

11.  A  boy  sells  newspapers  at  5  cents,  which  is  66|% 
above  cost;  find  the  cost. 

12.  A  boy  buys  pencils  for  25  cents  a  dozen,  and  sells 
them  for  5  cents  apiece;  what  is  his  gain  %  ? 

13.  Bought  oranges  for  1  cent  apiece,  an-d  sold  them  at 
the  rate  of  2  for  3  cents;  what  was  the  rate  of  gain? 

14.  Bought  4  books  for  $2.40,  lost  one,  and  sold  the  re- 
mainder at  $1  each;  find  my  gain  %. 

15.  A  man  bought  a  hat  for  $4,  and  traded  it  for  $3  and 
a  box  of  6  collars,  worth  25  cents  each;  wdiat  was  his  rate 
of  gain  ? 

16.  A  furniture  dealer  bought  a  second-hand  set  of  chairs 
at  32  cents  each,  spent  8  cents  each  in  repairs,  and  then 
sold  them  at  a  gain  of  25%;  what  did  he  receive  for  them? 

17.  Bought  a  suit  for  $25,  which  was  16|%  less  tlian  the 
asking  price,  and  the  asking  price  was  50%  above  the  cost; 
find  the  cost. 


ARITHMETIC.  189 

18.  20%  of  my  sales  is  profit;  what  is  my  gain  %? 

19.  Sold  4-  of  my  stock  for  what  the  whole  cost;  what  did 

I  gain  per  cent? 

EXERCISE  216. 

Find  everything  not  given;  gain  or  loss,  rate  of  gain  or 
loss,  cost,  and  selling  price. 

Cost  $1500;  loss  1\%.  13.  Loss  $125;  S.  P.  83^%. 

Gain  $500  at  2h^%.  14.  S.  P.  $480  or  73^%. 

S.  P.  $1320;  loss  \%.  15.  Cost  $920;  loss  $15%. 

Loss  $75;  cost  $2000.  16.  Loss  13^%=$840. 

5.  Gain  7f%;  cost  $1085.  17.  Gain  $5.50;  S.  P.  $95.50. 

6.  Cost  $2375;  S.  P.  $3050.  18.  Cost  $175;  S.  P.  $200. 

7.  Gain  1%  or  $147.  19.  Gain  14%;  cost  $7000. 

8.  Loss  16|%;  S.  P.  $2085.  20.  Profit  50%;  gain  $25.50. 

9.  Cost  $12.50;  S.  P.  $10.  21.  S.  P.  $175;  cost  $150. 

10.  S.  P.  $18.50;  loss  6^%.        22.  Cost  $15;  loss  20%. 

11.  Profit  $45  or  3^%.  23.  S.  P.  $15;  loss  16f  %. 

12.  Cost  $1300;  S.  P.  130%.     24.  Gain  3%;  S.  P.  $1030. 

EXERCISE   217. 

Select  10  examples  from  Exercise  209  and  form  practical 
examples  in  P.  and  L.  Perform  and  bring  to  the  class  for 
dictation. 

Model :  Example  1.     I  gained  $75  by  selling  goods  at  3%  profit ; 
what  did  thev  cost? 


COMMISSlOjSr. 

Some  men  are  employed  in  transacting  business  for  oth- 
ers, such  as  buying  and  selling  goods  or  lands,  renting 
houses,  collecting  money. 

These  men  are  found  in  our  business  centers  under 
various  names,  including  commission  merchants,  brokers, 
auctioneers,  real  estate  agents,  collectors,  and  the  like. 


190  CALIFORNIA   SERIES. 

The  money  received  for  their  services  is  called  Commis- 
sion. 

It  is  usually  a  percentage  (1)  on  the  money  paid  for  prop- 
erty bought,  (2)  received  for  property  sold,  (3)  on  the 
amount  of  money  collected. 

The  sum  left  after  taking  out  the  commission  is  called 
the  proceeds. 

A  broker's  commission  is  called  brokerage. 

r  Cost. 

Base=  -l  Selling  price. 

[  Money  collected. 

EXERCISE    218.    (Written.) 

Label  numbers  given  in  the  first  10  examples  with  the 
words  cost,  S.  P.,  money  collected,  proceeds,  commission,  or 
rate  of  commission  ;  rewrite  in  general  form,  said  perform. 

1.  I  send  50  tons  of  baled  hay  to  a  commission  merchant 
in  San  Francisco,  who  sells  it  for  me  at  $14  a  ton  and 
charges  S%  commission;  what  is  the  amount  of  his  com- 
mission, and  what  do  I  receive? 

Model:  50x$14  =  $700  S.  P.     3%  =  Rate  of  commission. 
What  is  3%  of  $700? 
What  is  97%  of  $700? 

2.  If  I  pay  $10.50  per  ton  for  the  hay  and  $1.14  per  ton 
for  freight,  what  %  do  I  make  on  the  whole  cost? 

3.  A  merchant  buys  100  barrels  of  flour  for  me,  paying 
$5.50  per  barrel.  If  he  charges  o%  commission  what  sum 
of  money  must  I  send  him  to  pay  for  the  flour  and  his  ser- 
vices? 

4.  I  send  $3120  to  a  commission  merchant  to  buy  flour 
at  4%  commission;  for  what  does  the  $3120  pay?  What 
is  the  cost  of  the  flour?     Commission? 

5.  For  what  must  I  sell  the  above  flour  per  barrel  to  gain 
205^  on  the  whole  cost,  supposing  I  received  750  barrels? 

6.  A  farmer  sends  72  dozen  eggs  to  an  agent  who  sells 


ARITHMETIC.  191 

them  @  32  cents  at  a  commission  of  ^\%]  what  does  the 
farmer  receive  for  his  eggs  per  dozen  ? 

7.  An  auctioneer  sells  at  auction  a  farm,  buildings,  stock, 
and  tools.  He  receives  $14000  for  the  farm  and  buildings, 
$2700  for  the  stock,  and  $1300  for  the  tools;  what  is  his 
commission  at  \\%  ? 

8.  A  man  sends  $31500  to  a  broker  to  buy  cotton  at  5X 
commission;  how  many  bales  at  $100  each  does  he  buy? 

9.  A  grocer  sends  $2490  to  a  commission  merchant  to 
buy  sugar  at  3f  %  commission.  If  he  pays  8  cents  a  pound 
for  the  sugar,  for  what  must  the  grocer  sell  the  whole  to 
gain  16f  %  on  the  whole  cost,  and  at  how  much  per  pound? 

10.  An  agent  sells  Blaine's  "Twenty  Years  in  Congress," 
at  $5  a  volume,  receiving  35%  commission;  how  many  vol- 
umes must  he  sell  to  make  $1400? 

11.  A  collector  collected  rents  at  3%  commission  and 
received  $87.60  for  his  services;  what  sum  of  money  did 
he  collect? 

12.  A  farmer  sends  3000  centals  of  wheat  to  a  commis- 
sion merchant  in  San  Francisco,  who  sells  it  at  $1.16f  per 
cental  at  a  commission  of  21%;  what  is  his  commission 
and  what  does  the  farmer  receive  for  his  wheat? 

13.  My  commission  for  selling  flour  for  $5150  is  $128.75; 
what  X? 

14.  T  sent  $5115  to  an  agent  who  buys  goods  for  a  com- 
mission of  $165;  what  %1 

15.  My  agent  received  $123  for  collecting  rents  at  2>%] 
how  much  money  did  he  collect? 

16.  I  pay  $275  for  a  house  lot  and  build  on  it  a  house 
costing  $1720,  which  my  agent  rents  for  $25  a  month, 
charging  b%  commission;  what  per  cent  do  I  make  a  year 
on  the  money  laid  out? 

17.  A  lawyer  collects  lh%  of  a  bill  of  $5600  and  charges 
6|%  for  collecting;  what  is  his  commission  and  what  does 
the  creditor  receive? 


192 


CALIFORNIA   SERIES. 


18.  A  town  owes  a  debt  of  $1890  which  is  to  be  collected 
from  the  people  of  the  town.  If  the  collector  charges  10% 
for  collecting,  what  sum  must  be  collected  to  pay  the  debt? 

19.  I  wish  to  gain  2h%  on  cloth  for  which  I  paid  $1.20 
per  yard,  b%  commission  to  my  agent,  and  1\  cents  per 
yard  for  freight;  what  must  be  the  selling  price? 

20.  I  send  $2689.75  to  my  agent  to  buy  pork  at  1\% 
commission;  how  many  pounds  can  he  buy  at  3 J  cents  a 
pound  ? 

21.  How  many  pounds  of  sugar  at  8^  cents  does  an  agent 
purchase  for  me,  if  his  commission  at  ol%  amounts  to 
$25?     What  does  the  sugar  cost  me  per  pound? 

22.  How  many  barrels  of  flour  at  $5  can  a  commission 
merchant  purchase  with  $5150  on  a  commission  of  3%? 

23.  Find  the  commission  on  the  sale  of  100  bales  of  cot- 
ton, averaging  480  lb.  to  a  bale,  at  $18  per  cwi,  the  com- 
mission being  h%. 

24.  An  agent  sells  450  tons  of  hay  at  $13  a  ton,  commis- 
sion 5X,  and  with  the  proceeds  bought  wool  at  22^  cents 
per  pound,  commission  4%;  what  was  his  whole  commis- 
sion and  how  many  pounds  of  wool  did  he  buy? 

25.  Bought  500  boxes  of  oranges  at  $2.50  a  box,  and  paid 
$12  freight.  My  whole  bill  was  $1287;  what  %  commission 
did  I  pay  for  buying  ? 


EXERCISE  219.    (Oral.) 

1.  My  agent  sells  $250  worth  of  goods  for  me  at  AX  com- 
mission; what  do  I  receive? 

2.  I  send  an  agent  sufficient  money  to  buy  $75  worth  of 
shoes  at  A%\  what  do  I  send  him? 

3.  A  commission  merchant  sold  a  bill  of  goods  at  3% 
commission,  receiving  $30  for  his  services;  what  was  the 
value  of  the  goods  sold? 

4.  A  commission  merchant  sells  goods  for  me  for  $200, 
receiving  $4  commission;  what  %? 


ARITHMETIC.  193 

v"  ) * 

5.  An  agent  receives  $2  for  buying  eggs  on  a  commission 
of  2% ;  what  does  he  pay  for  the  eggs  ? 

6.  An  auctioneer  sells  a  sewing  machine  for  $20,  receiv- 
ing 5/0  for  his  services;  what  is  the  sum  received  by  the 
owner  ? 

7.  If  a  commission  merchant  sells  flour  for  $5  a  barrel 
on  a  commission  of  5%,  how  many  barrels  must  he  sell  to 
realize  $100? 

8.  $10  commission;   10^  rate;  find  the  cost. 

9.  Bought  a  lot  of  clocks  through  an  agent,  pajdng  $50 
for  the  clocks  and  $2  commission;  what  was  the  rate  of 
commission  ? 

10.  For  how  much  a  yard  must  cloth  be  sold  to  gain 
33^%,  if  the  cloth  was  bought  @  20  cents  on  a  commission 
of  0%  ? 

11.  Sent  $30  to  an  agent  to  buy  lead  pencils  at  50^ 
commission;  how  many  at  2  cents  apiece  can  he  get? 

12.  Sales  $2000;  commission  $10;  find  the  rate. 

13.  What  amount  of  money  must  I  send  my  agent  that 
he  may  buy  100  pr.  of  shoes  at  $1  and  pay  himself  a  com- 
mission of  3%  ? 

14.  Remittance  $2020;  commission  IX;  find  cost.     ^ 

15.  Cost  $300;  remittance  $309;  find  commission  %. 

EXERCISE  220. 

Find  everything  not  given  of  the  following;  rate  of  com- 
mission, commission,  cost,  S.  P.,  proceeds. 

1.  Com.  $165;   (S.  P.)  $6600. 

2.  Com.  for  buying  $140  at  H%. 

3.  Remittance  to  agent  $5600;  com.  2\%. 

4.  Com.  for  selHng  at  \l%  is  $13.50. 

5.  Auction  sale  $8732;  com.  2%. 

6.  Com.  $14.21;  cost  $568.38. 

7.  Sum  collected  $14000;  com.  $420. 

8.  Com.  $141;  proceeds  $2209. 

13— A 


194 


CALIFORNIA   SERIES. 


9.  Remittance  to  agent  $4000;  com.  $250. 

10.  Remittance  to  agent  $2182.80;  com.  1%. 

11.  Cost  $4800;  remittance  $4872. 

12.  Com.  for  selling  $48.29  at  2|%. 

13.  Remittance  to  agent  $1500;  com.  $41.12. 

14.  Proceeds  $4975;  com.  $25. 

15.  Com.  for  buying  $74.25  at  H%. 

16.  Rate  of  com.  l%]  S.  P.  $2000. 

EXERCISE   221. 

Select  10  examples  from  Exercise  209  and  construct 
practical  examples  in  commission.  Perform  and  bring  to 
the  class  for  dictation. 


l^SUEAlsrOE. 

I  build  a  house  for  $3000.  A  company  gives  me  a  writ- 
ten promise  to  pay  me  $2000  if  the  house  burns,  and  charges 
me  ^%  per  year  of  the  promised  sum  for  the  promise.  What 
do  I  pay  for  the  promise?  What  does  the  company  lose  if 
the  house  burns  within  a  year  ? 

A  company  is  made  up  of  two  or  more  persons  joining 
for  the  transaction  of  business. 

Insurance  is  a  guaranty  of  a  sum  of  money  to  be  paid 
in  case  of  loss  of  property  or  life. 

The  company  making  the  written  contract  to  pay  losses 
is  an  Insurance  Company. 

The  written  contract  is  called  the  policy. 

The  sum  paid  by  the  owners  of  property  for  insurance  is 
called  the  premium. 

The  'premium  is  a  certain  fraction,  or  per  cent,  of  the  sum 
insured,  and  is  paid  in  advance. 

Fire  Insurance  Companies  rarely  insure  property  for  more 
than  f  of  its  value,  and  in  no  case  pay  for  more  than  the 


ARITHMETIC.  195 

value  of  the  property  destroyed,  whatever  may  be  the  face 
of  the  poHcy. 

In  Life  Insurance  the  premium  is  a  sum  of  money  vary- 
ing with  the  amount  of  insurance  and  the  age  of  the 
individual. 

A  fee  of  $1  or  more  is  sometimes  charged  for  making  out 
the  policy. 

Do  insurance  companies  lose  money  by  'paying  these  losses? 
Whyf 

EXERCISE  222. 

1.  A  merchant  has  his  store  and  contents  insured  for 
$5500  at  i%  premium;  what  is  the  cost  to  him?  If  the 
store  and  contents  are  destroyed,  what  sum  does  the  insur- 
ance company  lose? 

2.  A  trader  paid  $110  premium  to  have  a  shipment  of 
horses  insured  at  2|%  of  their  value;  what  was  their  value? 

3.  A  sea  captain  insures  his  vessel  for  $48000,  paying 
$360;  what  is  the  rate  of  insurance? 

4.  A  farmer  has  his  standing  grain,  worth  $4000,  insured 
for  ^  its  value  at  ^%  per  month.  At  the  end  of  a  month 
and  a  half  the  grain  burns;  what  does  the  insurance  com- 
pany lose? 

5.  I  pay  $62.50  to  insure  my  house  f(ti'  f  its  value  for  3 
years  at  2^%.     What  is  the  value  of  my  house? 

6.  An  insurance  company  loses  $3528  by  the  wreck  of  a 
carload  of  flour  which  it  had  insured  for  $3600.  What  was 
the  rate  of  insurance  ? 

7.  A  man  has  a  policy  of  $7600  placed  on  his  house, 
which  sum  includes  the  insurance  value  and  the  premium 
at  li%;  what  is  the  premium,  and  the  value  of  the  house? 

8.  I  have  a  house  worth  $6000,  a  barn  worth  $1800,  and 
personal  property  worth  $1200;  on  all  which  I  am  insured 
for  f  value,  paying  $106  including  $1  fee  for  the  policy. 
What  is  my  rate  of  insurance? 

9.  I  buy  a  house  for  $6500,  expend  $500  in  repairs,  and 


196  CALIFORNIA   SERIES. 

insure  it  at  ^%  on  f  of  the  whole  cost  including  repairs.  I 
then  sell  it  at  a  loss  of  4%  on  my  whole  expense.  What 
is  my  selling  price  ? 

10.  Sent  $2846.25  to  my  agent,  who  buys  flour  at  $5-1  a 
barrel,  charging  S^%  commission.  I  insure  it  at  !{%  on 
the  cost;  for  how  much  must  I  sell  it  per  barrel  to  gain 
10^  on  the  whole  cost? 

11.  A  house,  insured  for  $2400,  at  1%,  burns.  The  owner 
buys  another  with  the  insurance  money  and  gets  it  insured 
for  $30;  find  the  rate  on  the  latter  house. 

12.  Which  is  cheaper,  to  get  my  building  insured  in  two 
different  companies  for  $1500  each,  at  f%,  or  in  one  com- 
pany for  $3200  at  i%? 

13.  Paid  S%  every  3  years  to  get  an  insurance  of  $2400 
on  my  house.  If  it  burns  at  the  end  of  8  years,  what  is 
the  loss  of  the  insurance  company  ? 

14.  A  merchant  imports  a  cargo  from  Liverpool,  En- 
gland, worth  £1500  and  insures  it  at  ^%;  find  the  premium 
in  $'s. 

15.  Paid  $42  to  get  an  insurance  of  $2562  on  my  stock, 
the  insurance  covering  the  premium;  what  was  the  rate? 

16.  I  insure  my  life  for  $8000,  paying  $19.80  per  $1000 
per  year;  what  do  I  pay  the  company  if  I  live  20  years 
after  insurance  ? 

17.  Paid  li%  to  get  my  library  insured;  premium  6|X; 
what  was  the  value  of  the  library? 

18.  A  grocer  insures  200  barrels  of  flour  for  66f  %  of 
their  cost  at  1^%,  paying  a  premium  of  $10.50;  what  price 
per  barrel  must  the  flour  bring  to  gain  16|%  on  the  cost 
exclusive  of  insurance? 

19.  For  what  sum  must  a  policy  be  made  out  to  cover 
the  insurance  on  a  property  of  $2100,  at  ^%  ? 

20.  I  buy  a  house  for  $6000  and  pay  li%  to  get  it  insured. 
I  also  pay  $900  for  repairs.  I  then  sell  it  at  a  gain  of  oS^% 
on  all  it  has  cost  me.     The  money  thus  received  I  send  to 


ARITHMETIC.  197 

a  commission  merchant,  who  buys  flour  for  me  at  o^%  com- 
mission, paying  $4.50  per  barrel.  He  finally  sells  the  flour 
for  $4  a  barrel  at  3^%  commission  and  sends  the  balance 
to  me.  What  %  have  I  gained  on  all  I  paid  out  for  the 
house? 

EXERCISE  223.    (Oral.) 

1.  Paid  $7  on  an  insurance  of  $1400;  find  the  rate. 

2.  Paid  $20  to  get  stock  insured  for  |%;  find  the  value 
insured. 

3.  If  I  pay  H%  a  year  to  get  my  house  insured  for  $1500, 
how  many  years  will  it  take  to  pay  its  value  in  premiums? 

4.  Paid  $3  on  an  insurance  of  $400;  find  the  rate. 

5.  Paid  $12  to  insure  merchandise  at  li/1):  find  its  value. 

6.  Insared  \  share  in  a  ship  worth  $100000  for  \  value 
at  \%\  find  the  premium. 

7.  Gained  25^  by  selling  flour  for  $505,  the  cost  includ- 
ing an  insurance  of  1%  ;  find  the  first  cost. 

8.  I  insure  my  house  for  f  of  its  value  of  $3000,  and  my 
stock  for  f  of  its  value  of  $600,  for  ^%  premium;  find  the 
premium. 

9.  Paid  $2.50  on  an  insurance  of  $500;  find  the  rate. 

10.  Paid  1%  on  an  insurance  of  $900;  find  the  premium. 

11.  Paid  $5  to  get  an  insurance  at  2\%]  find  the  insur- 
ance. 

12.  Insured  standing  grain  worth  $2000  for  \  its  value  at 
\%  a  month;  what  do  I  pay  for  2  months'  insurance? 

13.  Paid  ^%  a  year  for  5  years  on  property  insured  for 
$5000;  if  it  burns  then,  what  is  the  company's  loss? 

14.  Paid  $13,  including  $1  for  policy,  on  an  insurance  of 
$1200;  what  w^^s  the  rate? 

15.  $1010  policy  including  1%  premium;  find  the  value. 

EXERCISE  224.    (Written.) 

Construct  10  examples  of  3"our  own  from  Exercise  209, 
perform,  and  bring  to  the  class  for  dictation. 


198  CALIFORNIA   SERIES. 


TAXES. 

Towns,  counties,  and  states  are  at  expense  to  maintain 
schools,  courts,  roads,  public  buildings,  officers,  and  the 
like.  To  meet  these  expenses  the  people  are  required  to 
pay  a  per  cent  of  the  value  of  their  property. 

The  money  paid  by  an  individual  for  public  expense  is 
called  a  Tax. 

Taxable  property  is  of  two  kinds:  (1)  Personal,  or  mov- 
able property;  as,  money,  tools,  carriages,  stock;  (2)  Real 
estate,  or  immovable  property;  as  lands,  buildings. 

In  California,  men  between  the  ages  of  21  and  60  are 
also  taxed  so  much  a  head  without  regard  to  property. 
This  is  called  poll  tax.  The  amount  raised  by  poll  tax 
makes  the  property  tax  so  much  less. 

EXERCISE  225.    (Written.) 

1.  Suppose  the  property  of  this  county  to  be  valued  at 
$4000000,  and  its  expenses  for  this  year  to  be  $21800.  If 
1200  men  pay  a  $1.50  poll  tax,  what  will  be  the  fraction, 
and  %,  of  tax  on  the  property? 

2.  If  your  parents  own  real  estate  valued  at  $4000  and 
personal  property  valued  at  $1800,  and  pay  1  poll  tax,  what 
is  their  whole  tax? 

3.  If  your  next  door  neighbor  pays  a  tax  of  $16,  includ- 
ing 1  poll,  what  is  his  property  valued  at? 

4.  Suppose  Mr.  B  pays  3  polls  and  has  real  estate  valued 
at  $5500  and  personal  property  valued  at  $1700;  what  is 
his  tax? 

5.  A  county  builds  a  bridge  for  $4500.  The  property  is 
valued  at  $1000000:  what  is  the  tax  per  $100? 

6.  A  tax  of  $8500  was  raised  on  a  town  at  $16  on  a  $1000 
worth  of  property.  If  there  were  500  polls  at  $1  each,  what 
was  the  value  of  the  town  property? 

7.  A  school  district  is   taxed   $3000   to  build  a  school 


ARITHMETIC.  199 

house,  which  sum  is  a  tax  of  -f^X  of  the  property  value; 
what  is  the  property  value  and  what  is  the  tax  on  a  dollar? 

8.  The  road  tax  on  a  road  district  was  2  mills  on  a  dol- 
lar; what  was  the  rate  per  cent  of  tax,  and  the  amount  on 
$3500  worth  of  property  ? 

9.  A  town  whose  property  value  is  $450000  has  an  ex- 
pense of  $4750.  If  a  collector  charges  o%  for  collecting, 
what  rate  of  taxation  must  be  made  ? 

10.  At  the  rate  of  8  mills  on  $1,  and  $2  poll  tax,  find  a 
man's  tax  on  $7500  real  estate,  $2750  personal  property, 
and  2  polls. 

11.  A  poll  tax  of  $2  for  road  improvements  is  assessed 
on  a  town  of  485  polls;  105^  is  paid  for  collecting.  What 
sum  of  money  will  be  left  for  improving  roads  ? 

12.  I  buy  a  house  lot  for  $400  and  build  a  house  on  it 
for  $2000.  I  pay  an  insurance  on  the  house  of  ^%  on  ^  its 
value,  and  a  tax  on  the  whole  of  $14  on  $1000,  the  property 
valuation  being  |  the  cost.  For  how  much  must  I  rent  the 
house  per  month  to  realize  20"^  a  year  on  my  money? 

13.  A  tax  of  $2850  is  to  be  raised  on  a  town  and  suffi- 
cient besides  to  pay  for  collecting  at  h%.  If  the  rate  is  \ 
cent  on  a  dollar,  what  is  the  property  worth  ? 

14.  I  buy  a  house  for  $6500  and  spend  $500  for  repairs. 
I  rent  it  for  $77.50  a  month,  out  of  which  I  pay  a  yearly 
insurance  of  ^%  on  f  of  its  whole  cost,  including  repairs, 
and  a  yearly  tax  of  \%  on  f  of  the  same.  What  per  cent 
of  income  a  year  do  I  realize  on  the  whole  cost? 

EXERCISE   226.    (Written.) 

Ask  your  parents  or  guardian  for  their  last  tax  bill. 
Bring  to  the  class  for  dictation. 

DUTIES. 

The  expenses  of  the  U.  S.  Government  are  mostly  paid 
by  taxes  on  imported  goods. 


200  CALIFORNIA   SERIES. 

Such  taxes  are  called  customs  or  duties. 

Another  object  of  duties  is  "  protection  to  home  industry." 
Find  out  what  you  can  about  "  j^rotection." 

Duties  are  of  two  kinds:   specific  and  ad  valorem. 

A  specific  duty  is  a  charge  on  goods  by  weight,  number, 
or  measure  without  regard  to  value. 

An  ad  valorem  duty  is  a  per  cent  of  the  cost  of  the  goods 
at  the  port  from  which  shipped.  Both  classes  of  duties 
are  laid  upon  some  goods. 

Gross  weight  is  the  weight  of  goods  including  the  boxes 
or  other  packing  material. 

Net  weight  is  the  weight  after  deducting  the  weight  of 
the  packing  material. 

Duties  are  estimated  on  the  net  weight,  and  all  custom- 
house weights  are  long  ton  weights. 

EXERCISE    227.    (Written.) 

State  in  connection  with  each  example  whether  the  duty 
is  specific  or  ad  valorem. 

1.  Find  the  duty  on  100  boxes  of  oranges  at  25  ct.  per  box 
and  60  boxes  of  lemons  at  30  ct.  per  box. 

2.  What  is  the  duty  on  100  French  watches  valued  at 
$15  each,  duty  25%? 

3.  Imported  11  tons  of  iron  T  rails,  duty  y%  ct.  ^  ib. 
What  was  the  whole  duty? 

4.  A  merchant  imported  12  cases  woolen  shawls,  each 
case  averaging  255  ib.  valued  at  80  ct.  ^  ib.,  duty  35  ct.  ^  ib. 
and  35%  ad  valorem.    Charges  $72.50.    Find  the  whole  cost. 

5.  A  liquor  dealer  imported  80  doz.  quart  bottles  of 
champagne,  duty  $7  per  doz.  bottles;  3  casks  of  French 
brandy,  30  gal.  each,  duty  $2  per  gal.;  3  casks  wine,  31^ 
gal.  each,  duty  50  ct.  per  gal.;  and  50  doz.  pint  bottles  of  ale, 
duty  35  ct.  per  gal.     Find  the  whole  duty. 

6.  Paid  a  duty  of  $2283.60  on  an  invoice  of  silks  at  60% 
ad  valorem.     What  was  the  value  of  the  goods? 


ARITHMETIC.  201 

7.  Find  the  duty  on  1280  sq.  yd.  Brussels  carpet  valued 
at  $725,  duty  30  ct.  per  sq.  yd.  and  oO%  ad  valorem;  1440 
sq.  yd.  tapestry  valued  at  $G50,  duty  20 ct.  per  sq.  yd.  and 
30X  ad  valorem. 

8.  Duty  on  840  ft.  flaxseed  at  20  ct.  per  bu.  of  56  ft. 

9.  A  coal  dealer  imported  from  Sydney,  Australia,  1000 
tons  of  coal,  paying  75  ct.  per  ton  duty.  What  was  the  cost 
of  importing? 

10.  A  ship  brought  into  port  200  tons  of  rock  salt.  What 
was  the  duty  at  8  ct.  per  cwt.  ? 

11.  Imported  50  boxes  tin  plate,  108  ft.  to  the  box  net, 
on  which  I  paid  a  duty  of  1  ct.  per  ft.  W^hat  did  the  duty 
amount  to? 

12.  A  merchant  imported  25  tons  of  coke  invoiced  at  $94, 
duty  20%.     Find  the  duty. 

13.  Find  the  duty  at  20%  on  an  importation  of  Bath 
brick  of  200  boxes  valued  at  45  ct.  per  box. 


STOCKS. 

AVhen  a  number  of  men  wish  to  form  a  railroad  company, 
insurance  company,  bank,  or  the  like,  they  obtain  permis- 
sion by  law,  and  subscribe  a  sum  of  money  for  the  under- 
taking. 

Companies  authorized  by  law  to  carry  on  business  are 
called  corporations. 

The  money  invested  is  called  the  Stock. 

The  stock  is  divided  into  equal  parts,  commonly  of  $100 
each,  called  shares.  It  may  be  bought  and  sold  in  the 
market  like  other  property,  its  value  depending  mainly 
upon  the  prosperity  of  the  company. 

The  nominal  value  of  the  stock  is  called  the  par  value. 

The  price  which  the  stock  brings  in  the  market  is  called 
the  market  value. 


202  CALIFORNIA   SERIES. 

If  the  market  value  is  above  par,  the  stock  is  at  a  pre- 
mium; if  below,  at  a  discount.  Thus,  stock  selling  for  $118 
per  share  is  at  18%  premium;  for  $82,  at  18%  discount. 

Usually,  the  broker's  commission  for  buying  or  selling  is 
a  per  cent  of  the  jxrr  value  of  the  stock  dealt  in;  but  in 
mining  stocks  it  is  reckoned  upon  the  market  value  of  the 
stock  bought  or  sold. 

The  earnings  of  the  company,  after  deducting  the  ex- 
penses, are  divided  among  the  stockholders,  and  are  called 
dividends. 

Dividends,  premium,  and  discount  are  reckoned  on  the 
par  value. 

The  following  table  is  taken  from  the  -stock  quotations 
(market  value)  found  in  the  daily  papers,  par  value  $100: 

N.  Y.  Central  R.  R.,    ....  if;i02  Western  Union  Tel.,    .    .    .  $66| 

Mich.  Central  R.  R. 70  Home  Mut.  Ins., 145 

Lake  Shore  R.  R 81|  Bank  California, 169^ 

St.  Paul  R.  R., 87^  First  National, 125 

So.  Pacific  R.  R., 109^  Spring  Valley  Water,    .     .     .    91J 

Geary  St.  R., 107  Oregon  Navigation,  ....    99 

Giant  Powder, 60  Bodie  Mining, 1^ 

\Vells,  Fargo  Ex., 118  Mono  Mining, 2| 

EXERCISE  228.    (Written.) 

1.  Make  separate  lists,  from  the  above,  of  stocks  at  a  pre- 
mium and  those  at  a  discount. 

2.  What  must  I  pay  for  10  shares  of  N.  Y.  Central,  bro- 
kerage i%? 

3.  After  buying  the  above  stock,  a  dividend  of  4%  was 
declared;  what  rate  per  cent  of  income  did  I  realize  on  the 
money  invested? 

4.  A  friend  received  $320  in  dividends  at  the  same  time; 
how  many  shares  did  he  own,  and  what  w^ere  they  worth  at 
the  market  value? 

5.  The  Giant  Powder  Co.  declares  an  annual  dividend  of 
9% ;  how  many  shares  at  the  above  quotation  must  I  buy  to 
get  an  annual  income  of  $720,  and  what  will  they  cost  me? 


ARITHMETIC,  203 

6.  What  per  cent  on  investment  is  realized  by  a  dividend 
of  1%  in  the  Home  Mutual  if  i%  is  paid  for  brokerage? 

7.  Which  is  the  better  investment:  Bank  Cal.,  paying 
12%  dividends,  or  First  National,  paying  9%? 

8.  I  bought  a  certain  number  of  shares  of  Lake  Shore  at 
81 1,  and  sold  them  at  par,  brokerage  \%  on  each  transac- 
tion, thereby  gaining  $2860  on  the  whole;  how  many  shares 
were  there,  and  what  did  they  cost  me? 

9.  Find  my  gain  %  in  the  preceding  example. 

10.  Bought  a  house  for  $5000,  and  rented  it  for  a  year  for 
$275;  out  of  the  rent  paid  taxes  at  the  rate  of  1%  on  f  cost. 
At  the  end  of  the  year  T  sold  the  house  for  \2%  advance  on 
cost,  and  invested  the  sum  in  Michigan  Central,  paying 
^\%  dividends.     Which  was  the  better  investment? 

11.  I  send  a  broker  $1468  to  buy  Sp.  V.  W.,  at  1%  com- 
mission; how  many  shares  can  he  buy  me? 

12.  What  cost  150  shares  Mono  at  \%  brokerage? 

13.  What  dividend  would  have  to  be  declared  to  realize 
11-^ >6  on  money  invested  in  Oregon  Nav.  ? 

14.  What  is  the  cost  of  50  shares  of  ^lono  and  100  shares 
of  Bodie,  brokerage  \%  ? 

15.  Lost  $340  by  buying  S.  P.  R.  R.  at  the  above  quota- 
tion and  selling  at  101|,  brokerage  \%  each  way;  how 
many  shares? 

16.  How  many  shares  of  Western  Union  must  I  own  to 
realize  $570  on  a  6%  dividend,  and  for  what  will  they  sell 
at  the  above  quotation  ? 

17.  Which  is  better,  St.  Paul  R.  R.,  paying  b%,  or  Ore- 
gon Nav.,  paying  6X? 

18.  How  many  shares  in  Geary  St.  R.  could  3'ou  buy  for 
the  money  you  receive  by  selling  24  shares  of  Electric  Light 
at  53^,  no  brokerage? 

19.  How  many  shares  of  Wells,  Fargo  can  I  buy  for 
$1182.50  at  1%  commission? 


204  CALIFORNIA   SERIFS. 


INTEEEST. 

Suppose  a  friend  of  yours  wishes  to  buy  a  house  for  $3000. 
Not  having  that  sum  in  ready  money,  he  borrows  $1000  of 
you  and  agrees  to  pay  you  S%  of  the  sum  per  year  so  long 
as  he  keeps  it.  What  does  he  pay  you  a  year  for  the  use 
of  the  money?  What  for  2  yr.?  If  he  keeps  it  for  1  year, 
what  is  the  whole  sum  of  money  he  pays  you  ?  If  he  keeps 
it  2  yr.  ?     3  yr.  ?     2^  yr.  ?     3  yr.  6  mo.  ?     2  yr.  3  mo.  ? 

Money  paid  for  the  use  of  money  is  called  Interest. 

The  sum  lent  is  called  the  principal. 

The  per  cent  of  interest  for  a  given  time  is  called  the 
rate.  It  is  understood  as  being  for  a  year  unless  otherwise 
specified. 

The  principal  and  interest  added  make  the  amount. 

Banks  usually  loan  money  by  the  month;  and  sometimes 
pay  on  deposits  from  3  to  5%  a  year.  They  reckon  12 
months  of  30  days  each,  or  360  days,  to  a  year. 

The  United  States  laws  count  365  days  to  a  year,  but 
this  reckoning  is  not  in  common  use  among  business  men. 

EXERCISE  229.    (Oral.) 

Find  the  interest  and  amount  of : 

1.  $100,  at  6%,  for  2  yr.;  3  yr.;  3^  yr. 

2.  $200,  at  b%,  for  4  yr.:  4  yr.  6  mo. 

3.  $300,  at  8%,  for  6  mo.;  1  mo.;  3  mo. 

4.  $250,  at  6%,  for  2  yr.;  2^  yr.;  2  yr.  4  mo. 

5.  $150,  at  4%,  for  2  mo.;  6  mo. 

6.  $1,  at  Q%,  for  1  yr.;  1  yr.  4  mo. 

7.  $2,  at  5%\  for  3^  yr.;  2^  yj. 

8.  $5,  at  8%,  for  3  mo.;  3  yr. 

9.  $10,  at  3%,  for  5  mo.;  7  mo. 

10.  $20,  at  6X,  for  9  mo.;  8  mo. 

11.  $2.40,  at  5%,  for  2  yr.;  2|  yr. 


ARITHMETIC.  205 

12.  $7,  at  8/0,  for  G  mo.;  for  2  3T. 

13.  $25,  at  A%,  for  2  yr.;  at  8%. 

14.  $50,  at  5X,  for  6  mo.;  at  Q>%. 

15.  $800,  at  3%,  for  1  mo.;  at  6X. 

16.  $800,  at  b%,  for  9  mo.;  at  10^. 

17.  $750,  for  2\  yr.,  at  A%-  at  8%. 

18.  $1000,  for  3  mo.,  at  4X;  at  5X;  at  6%. 

19.  $1000,  for  2  yr.  9  mo.,  at  A%-  at  6%. 

20.  $1500,  for  4  yr,,  at  5X;  at  6%;  at  8X. 

21.  $2000,  for  2yV  jr.,  at  b%]  at  4%;  at  6^. 

22.  $2500,  for  2  mo.,  at  \%  a  month. 

23.  $450,  for  3  mo.,  at  1%  a  month. 

24.  $40,  for  5  mo.,  at  \%  a  month. 

EXERCISE  230.   (Oral.) 
To  find  the  years,  months,  and  days  between  two  dates. 

1.  Find  the  time  from  January  16th,  1884,  to  May  27th, 
1886. 

Method :  Jan.  16th,  1884,  to  Jan.  16th,  1886,    ...     2  yr. 
Jan.  16th,  1886,  to  May  16th,  1886,  ...     4  mo. 
May  16th,  1886,  to  May  27th,  1886,  .     .     .  11  da. 
Ans. — 2  yr.  4  mo.  11  da. 

2.  Find  the  time  from  October  25th,  1885,  to  May  10th, 
1887. 

Oct.  25th,  1885,  to  Oct.  25th,  1886, 1  yr. 

Oct.  25th,  1886,  to  Apr.  25th,  1887, 6  mo. 

In  April  after  25th,  5  days,  and  10  days  in  May  =  15  da. 
Ans. — 1  yr.  6  mo.  15  da. 

3.  Find  the  time  from  each  date  except  the  last  to  all 
the  following  dates  in  this  list: 

January  7th,  1880;  May  3d,  1881;  August  25th,  1882; 

Sept.  4th,  1883;         Dec.  1st,  1884;  Dec.  27th,  1885. 

Find  the  time  between  January  3d,  1885,  and  each  of  the 
above  dates.  Also  from  August  7th,  1881,  to  each  of  the 
above  dates. 


206  CALIFORNIA    SERIES. 

Suggestion. — The  teacher  will  give  additional  examples  as  needed 
until  the  class  is  quick  in  the  work  of  finding  the  time,  using  tlie 
pencil  or  crayon  to  record  results  only. 


SIX  PER  CENT  METHOD. 
EXERCISE  231.  (Written.) 

6  hundredths  of  the  principal  per  year  means  half  as 
many  hundredths  as  months;  therefore  add  ^  the  number 
of  months  to  6  times  the  number  of  years  for  the  hun- 
dredths of  the  multiplier.  An  odd  month  gives  ^  hun- 
dredth, or  5  thousandths;  and  since  a  month,  or  30  days, 
gives  5  thousandths,  -J-  of  30  days,  or  6  days,  gives  1  thou- 
sandth.    Therefore, 

To  form  a  multiplier. 

Take  6  times  the  years  and  |  the  months  as  hundredths, 
and  ^  the*  days  as  thousandths. 

Example :  Find  the  interest  of  $275.75  for  2  yr.  5  mo. 
18  da.  at  Q%  yearly. 


WORK. 

2x6+1- 

-.14 

2  /  5.7  5      Principal. 

5+Y    = 

=.008 

.148  Multiplier, 

.148 

2758 

1103 

2  20 

$40.81       Interest. 

A  little  practice  will  enable  the  student  to  form  a  Q% 
multiplier  very  quickly.  A  good  arithmetician  will  find 
the  time  between  two  dates  and  form  a  Q)%  multiplier  from 
it  in  80  seconds  or  less.  The  contraction  in  multiplication, 
p.  112,  should  always  be  used. 

Form  6%  multipliers  from  each  of  the  differences  between 
dates  found  in  Example  3,  Exercise  230. 


ARITHMETIC. 
Fill  out  the  following  table,  rate  Q%: 


207 


No. 

Date. 

Date. 

Principal. 

Interest. 

Amoxint. 

1 

Aug.  4,  1881. 
March  19,  1879. 
July  8,  1883. 
Jan.  16,  1884. 
Oct.  28,  1885. 
Dec.  1,  1885. 
June  10,  1883. 
April  14,  1887. 

Sept.  12,  1882. 
February  25, 1882. 
Sept.  24,  1886. 
May  8,  1886. 
Jan.  12,  1886. 
March  12,  1887. 
Jan.  4,  1887. 
May  8,  1887. 

$179.50 
325.00 
758.75 

1024.25 

584.50 

725.84 

387.95 

42.20 

9 

9 

9 

9 

9 

3 

9 

9 

4 

9 

9 

5 

9 

9 

6 

9 

9 

7 

9 

? 

8 

9 

9 

EXERCISE  232.    (Written.) 
To  find  the  interest  at  other  rates  than  6  per  cent. 

First  find  the  interest  at  6%;  for  5%,  subtract  -J-  of  this 
interest  from  itself;  for  A%  subtract  -g. 

For  1%  add  i  of  the  6%  interest  to  itself;  for  8%  add  I; 
for  9%  add  -J;  for  10%  divide  by  6,  removing  the  decimal 
point  one  place  to  the  right. 

If  higher  rates  are  needed,  form  a  12%  multiplier  with 
the  months  as  hundredths,  and  I  the  days  as  thousandths. 

Find  the  interest  on: 

1.  $450,  from  Mar.  7,  1885,  to  July  7,  1885,  at  4%;  5%; 
6%. 

2.  $387,  from  May  3,  1884,  to  Aug.  3,  1886,  at  5%;  7%; 
8%. 

3.  $718.25,  from  Jan.  1,  1885,  to  Jan.  1,  1887,  at  4%. 

4.  $410,  for  3  vr.  3  mo.  10  da.,  at  7%. 

(3  mo.  10  da.=100  da.=^g=tVye'ir.) 

5.  $718,  from  May  11,  1882,  to  May  31,  1885,  at  5%. 

6.  $380,  from  February  10,  1883,  to  May  5,  1885,  at  7%. 

7.  $425,  for  2  yr.  5  mo.  17  da.,  at  8%. 

(Divide  interest  of  1  vr.  bv  12  to  get  int.  for  1  mo.    2  yr.  5  mo. 
17  da.=29H  nio.)^ 

8.  $910.50,  from  Jan.  1,  1885,  to  Mar.  15, 1885,  at  6%. 

9.  $748,  from  April  3,  1886,  to  Aug.  24,  1886,  at  5%. 


208  CALIFORNIA    SERIES. 

10.  $875,  from  July  7,  1886,  to  Jan.  1,  1887,  at  4%. 

11.  $2512,  from  May  1,  1884,  to  May  10,  1885,  at  7%. 

12.  $3850,  from  Mar.  9,  1885,  to  Sept.  9,  1885,  at  S%. 

EXERCISE  233.    (Written.) 
Find  the  interest  on: 

1.  $431,  for  3  yr.  2  mo.  12  da.,  at  6X;  at  7%. 

2.  $1515,  for  1  yr.  1  mo.  1  da.,  at  6%;  at  4%. 

3.  $495,  for  5  mo.  24  da.,  at  7^%. 

4.  $218.50,  for  1  yr.  3  mo.  15  da.,  at  4^%. 

5.  $729,  for  2  mo.  at  S%;  at  S%.  ' 

6.  $435,  for  4  mo.,  at  7%;  at  5%. 

7.  $760,  for  1  yr.  9  mo.  27  da.,  at  Q%;  at  S%. 

8.  $129.40,  for  7  mo.  16  da.,  at  4%;  at  5%. 

9.  $240.50,  for  19  mo.  18  da.,  at  7^%. 

10.  $528,  from  Jan.  1,  1884,  to  May  16,  1886,  at  4^%. 

11.  $1150,  from  Mar.  19,  1884,  to  July  25,  1884,  at  7%. 

12.  $1425,  from  May  3,  1885,  to  Sept.  30,  1886,  at  Q>%. 

13.  $45,  from  Aug.  7,  1885,  to  Jan.  13,  1886,  at  5%. 

14.  $75,  from  Apr.  28, 1884,  to  Apr.  10,  1885,  at  Q%. 

15.  $110,  from  May  23,  1880,  to  Sept.  13,  1884,  at  4%. 

16.  $434.20,  from  Dec.  1, 1881,  to  Nov.  1,  1884,  at  4^%. 

17.  $290,  for  1  yr.  11  mo.,  at  S^%. 

18.  $4050,  for  5  mo.  10  da.,  at  5%. 

19.  $1235,  from  May  19,  1886,  to  Sept.  1,  1886,  at  6%. 

20.  $1425,  from  Jan.  25, 1884,  to  May  10,  1885,  at  5%. 

21.  $475,  for  8  mo.  8.  da.,  at  S%. 

22.  $2150,  for  21  da.,  at  6%. 

23.  $1240,  for  17  da.,  at  4%. 

24.  $1345,  from  May  1,  to  May  25,  at  Q%. 

Work  the  examples  of  Exercise  229  by  this  method. 

EXERCISE  234.    (Written.) 
Find  the  amount  of : 

1.  $980,  for  7  mo.  10  da.,  at  6%. 

2.  $418.25,  for  3  mo.,  at  i%  per  mo. 


ARITHMETIC.  209 

3.  $7280,  from  Mar.  1  to  May  13,  at  1%  per  mo. 

4.  $1212.50,  for  1  yr.  1  mo.  14  da.,  at  ^%. 

5.  $976.10,  from  May  27  to  Nov.  19,  at  b%. 

6.  $3200,  for  9  mo.  9  da.,  at  ^%. 

7.  $225,  from  June  29  to  Dec.  1,  at  1\%. 

8.  $850,  from  Feb.  1  to  Sept.  1,  at  li%  per  mo. 

9.  $230,  for  1  mo.  10  da.,  at  10%. 

10.  $1925,  for  4  mo.  4  da.,  at  b%. 

11.  $458,  from  Jan.  1,  1887,  to  Mar.  11,  1888,  at  Q>1%. 

12.  $319.50,  for  3  mo.,  at  8%. 

13.  $112.75,  for  2  yr.  5  mo.  25  da.,  at  6%. 

14.  $550,  from  Apr.  3  to  Nov.  9,  at  b%. 

15.  $336,  from  Sept.  20,  1885,  to  Mar.  1,  1886,  at  6%. 

16.  $210,  for  2i  yr.,  at  1\%. 

17.  $640,  for  9^  mo.,  at  8%. 

18.  $1350,  from  Mar.  1  to  Sept.  1,  at  10%. 

19.  $2080,  for  4  mo.,  at  A\?/o. 

20.  $1875.35,  from  July  7  to  Jan.  1,  at  Q>1%. 

21.  $70,  for  11  mo.,  at  6X. 

22.  $10.50,  from  Jan.  1  to  July  10,  at  10^^. 

23.  $49.50,  for  1  yr.  7  mo.  28  da.,  at  A%. 

24.  $112,  for  2  yr.  12  da.,  at  Q>%. 

25.  $129.75,  for  2  yr.  17  da.,  at  b\%. 

26.  $18.50,  from  Jan.  1,  1808,  to  Aug.  17,  1887,  at  Z%. 

EXERCISE  235.    (Written.) 

Construct  10  examples  of  your  own,  find  the  amount  in 
each,  and  bring  to  the  class  for  dictation. 

To  compute  accurate  interest. 

When  interest  is  to  be  reckoned  on  a  basis  of  365  days 
to  a  year,  count  the  exact  number  of  years  and  days  be- 
tween the  dates.  Find  the  interest  for  years  as  in  the 
ordinary  method.     For  the  days  take  as  many  365ths  of  1 

year's  interest  as  there  are  days.     Thus, 
14— A 


210  CALIFORNIA  SERIES. 

Find  the  exact  interest  on  $240  from  March  1,  1885,  to 
July  10,  1885,  at  b%.     (131  da.) 

OPERATION. 

12 

EXERCISE  236.    (Written.) 
Find  the  exact  interest  on: 

1.  $219,  for  25  da.,  at  1%. 

2.  $480,  from  May  10,  1884,  to  July  3,  1886,  at  Q>%. 

3.  $348,  for  73  da.,  at  6^%. 

4.  $1000,  for  219  da.,  at  A%. 

5.  $1220,  from  March  27  to  July  27,  at  10%. 

6.  $104,  from  Jan.  9  to  Apr.  4,  at  12%. 

7.  $210,  from  Apr.  1,  1886,  to  July  12,  1887,  at  1\%. 

8.  $442,  for  91  da.,  at  b%. 

9.  $920,  from  Aug.  17  to  Dec.  1,  at  8%. 

10.  $460,  for  75  da.,  at  10%. 

11.  $235,  from  May  15,  1884,  to  July  27,  1886,  at  A%. 

12.  $40,  for  40  da.,  at  12%. 


PROBLEMS  IN  INTEREST. 
Analyze  by  model  under  Exercise  69: 

1.  At  7  per  cent,  $500  gains  $35  in  1  year;  how  many 
years  will  it  take  to  gain  $105? 

2.  ki  1%  $500  gains  $15  in  3  years;  at  how  many  per 
cent  will  it  gain  $105  in  the  same  time? 

3.  At  7  per  cent,  in  3  years  $1  gains  21  cents;  how  manyi 
dollars  will  it  take  to  gain  $105  at  the  same  rate  and  time? 

4.  At  7  per  cent,  in  3  years  $1  amounts  to  $1.21;  how] 
many  dollars  will  amount  to  $605  at  the  same  rate  andj 
time? 


I 


ARITHMETIC.  211 

In  Example  1,  by  knowing  the  rate  we  know  the  interest 
for  1  yr.  In  Example  2,  we  know,  without  stating,  tlie  in- 
terest at  1%  for  3  yr.  In  Example  3,  we  know  the  interest 
of  '$1  for  3  yr.  at  1%]  and  in  Example  4,  the  amount  of  the 
same.     Hence,  the  examples  may  be  contracted  thus: 

1.  In  what  time  will  $500  gain  $105,  at  7>o  ? 

2.  At  what  rate  will  $500  gain  $105  in  3  yr.? 

3.  What  sum  will  gain  $105  in  3  yr.,  at  7%? 

4.  What  sum  will  amount  to  $605  in  3  yr.,  at  7%? 

Observe  /  First  apply  the  co7iditions  of  the  examples  to  a 
\      unit,  or  1,  of  the  things  asked  for  in  the  answer. 

EXERCISE  237.    (Oral.) 
Find: 

1.  Time  in  which  $100  will  gain  $15,  at  Q%. 

2.  Sum  that  will  gain  $20  in  4  years,  at  b%. 

3.  Rate  at  which  $50  will  gain  $1.50  in  6  mo. 

4.  Sum  that  will  gain  $30  in  3  yr.,  at  b%. 

5.  Rate  at  which  $200  will  gain  $25  in  2^  yr. 

6.  Time  in  which  $75  will  gain  $5,  at  4%. 

7.  Rate  at  which  $60  will  gain  $7.50  in  2|  yr. 

8.  Time  in  wiiich  $150  will  gain  $21,  at  S%. 

9.  Sum  that  will  gain  $100  in  10  yr.,  at  10.9^. 

10.  Sum  that  will  amount  to  $12  in  2  yr.,  at  lOX. 

11.  Time  in  which  $1000  will  gain  $90,  at  4^%. 

12.  Rate  at  which  $800  will  gain  $40  in  1  yr.  3  mo. 

13.  Sum  that  will  gain  $75  in  5  yr.,  at  b%. 

14.  Rate  at  which  $300  will  gain  $28  in  2  yr.  4  mo. 

15.  Time  it  will  take  $700  to  amount  to  $749,  at  7%. 

16.  Rate  at  which  $75  gains  $4  in  8  mo. 

17.  Sum  that  gains  $200  in  2  yr.,  at  b%. 

18.  Rate  at  which  $450  gains  $72  in  2  yr.  8  mo. 

19.  At  what  rate  any  sum  will  double  itself  in  4  yr.;  8 
yr.;  10  yr. 

20.  Timeitwill  take  money  to  double  itself,  at  5%;  at  6%. 


212  CALIFORNIA   SERIES. 

EXERCISE  238.    (Written.) 

1.  Find  the  time  in  which  $360  will  gain  $97.20,  at  6%. 

2.  In  what  time  will  $900  gain  $84,  at  1%]  at  8%? 

3.  What  sum  will  gain  $62.50  in  2  yr.  6  mo.,  at  5%? 

4.  Find  rate  at  which  $145  will  gain  $5.80  in  6  mo. 

5.  Rate  at  which  $240  will  gain  $56  in  3  yr.  6  mo. 

6.  What  smn  will  amount  to  $296  in  3-|  yr.,  at  7%  ? 

7.  A  merchant  buys  goods  for  $700,  to  be  paid  in  6  mo.; 
what  sum  put  at  interest  to-day  at  6%  will  pay  the  debt? 

The  money,  which,  put  at  interest  at  the  present  time,  will 
amount  to  a  given  sum  in  a  given  time,  is  sometimes  called 
the  present  worth;  and  the  difference  between  the  present 
worth  and  amount,  the  true  discount. 

8.  How  long  will  it  take  $720  to  gain  $16.20  at  1\%  a  mo.  ? 

9.  Find  present  worth  of  $400  due  in  4  mo.,  at  \%  a 
month. 

10.  Find  true  discount  of  $390  in  6  mo.,  at  6%. 

11.  A  man  was  offered  a  horse  for  $100  cash,  or  $104  in 
6  mo. ;  if  money  is  worth  8%,  which  is  the  better  offer? 

12.  How  long  must  $450  be  kept  at  interest,  at  8,%,  to 
gain  what  $700  gains  in  2  yr.,  at  4j)6  ? 

13.  A  man  owes  3  bills  of  $250  each,  due  in  4,  6,  and  9 
months  respectively;  what  are  the  debts  worth  to-day,  at 
1%  a  month? 

14.  Bought  a  house  for  $7500,  payable  in  4  mo.,  and  sold 
it  for  $7500  cash;  if  money  is  worth  \%  a  month,  what  did 
I  gain? 

15.  A  house  that  cost  $3400  rents  for  $35  a  month,  what 
annual  rate  of  interest  is  received? 

16.  Find  rate  at  which  $275  will  gain  $56.10  in  3  yr.  4 
mo.  24  da. 

17.  Find  principal  that  will  gain  $103.95  in  3  yr.  2  mo. 
15  da.,  at  7^%. 

18.  What  sum  of  money  invested  at  6%  will  give  an 
income  of  $100  per  month? 


ARITHMETIC.  213 

19.  Find  principal  that  will  amount  to  $926.06,  at  6%,  in 
3  yr.  7  mo.  21  da. 

20.  Find  time  in  which  $720  will  amount  to  $736.20,  at 
1-|%  a  month. 

Note. — Find  the  interest  first. 

21.  Find  time  in  which  $125,  at  4%,  will  amount  to 
$141.50. 

22.  Find  rate  at  which  $760  will  amount  to  $926.06  in  3 
yr.  7  mo.  21  da. 

23.  Paid  a  debt  due  Apr.  1,  1886,  which  amounted  to 
$221.27  June  10,  at  6%:  find  the  debt. 

24.  I  loaned  my  money  at  S%,  payable  quarterly,  and 
received  $125  a  quarter.     How  much  did  I  loan? 

25.  What  principal  amounts  to  $560.23  in  2  yr.  7  mo.  15 
da.,  at  6%  ? 

26.  Borrowed  $90,  June  1,  1880,  at  7%.  Paid  it  when 
it  amounted  to  $100;  when  did  I  pay  it? 

27.  Paid  $71.30,  at  5%,  for  the  use  of  $460  how  long? 

28.  If  I  owe  $200  payable  in  2  mo.,  $300  in  3  mo.,  and 
$400  in  4  mo.,  what  should  I  pay  to-day  to  make  the  debt 
good,  money  being  worth  ^%  Si  month? 

29.  A  carriage  for  which  I  paid  $200  cash,  I  sold  for  $210 
on  8  mo.  credit.     Money  being  worth  Q%,  what  did  I  gain? 

30.  Find  rate  at  which  $410  gains  $27.06  in  1  yr.  1  mo. 
6  da. 

31.  Paid  in  4  yr.  $210  interest,  at  7%.  What  was  the 
principal  ? 

32.  Find  time  in  which  $550  will  gain  $102,  at  6%. 

33.  Find  difference  between  the  interest  and  true  discount 
of  $270  for  9  mo.,  at  S%. 

34.  Borrowed  a  sum  of  money  at  6%  and  lent  it  again  at 
7^%,  by  which  I  gained  $35.10  in  3  yr.  What  was  the  sum? 

35.  Find  rate  at  wdiich  $75  will  gain  $2  in  -^  of  a  year. 

36.  Find  present  vahie  of  $2000,  i  due  in  2  mo.,'  i  in  3 
mo.,  and  the  remainder  in  5  mo.,  at  6%, 


214  CALIFORNIA   SERIES. 

EXERCISE  239.    (Written.) 

Select  10  examples  from  Exercise  232,  perform,  and  then 
form  different  problems  in  interest  from  them,  and  bring 
to  the  class  for  dictation. 


PAETIAL  PAYMENTS. 

When  a  person  borrows  money  it  is  customary  to  give  the 
lender  a  written  promise  to  pay  it  back,  with  other  specifi- 
cations, as  that  of  interest,  stated.  Thus,  if  I  borrow  $500 
of  James  Willson  of  Sacramento,  at  7%,  I  write: 

$500.  Sacramento,  Cal.,  Aug.  8,  1885. 

Six  months  after  date,  value  received,  I  promise  to  pay 

James  Willson,  or  order,  Five  Hundred  -j^o  Dollars,  with 

interest  at  seven  per  cent  per  annum. 

Samuel  Jones. 

A  written  promise  to  pay  a  sum  of  money  is  called  a 
note.  The  date  at  which  the  money  is  to  be  paid  is  called 
its  maturity. 

A  note  containing  the  words  "or  bearer"  may  be  col- 
lected when  due  by  the  person  having  it  in  possession. 

If  James  Willson  wishes  to  make  the  above  note  payable 
to  bearer,  he  indorses  it  with  his  name.  If  he  wishes  to 
make  it  payable  to  Alfred  Smith  he  indorses  it: 

Pay  to  Alfred  Smith,  or  order. 

James  Willson. 

Alfred  Smith  may  transfer  it  in  the  same  way. 

One  who  indorses  a  note  becomes  responsible  for  its  pay- 
ment. 

The  face  of  a  note  or  other  business  paper  is  the  sum 
mentioned  in  it. 

If  Samuel  Jones  wishes  to  make  the  above  note  a  demand 


i 


ARITHMETIC.  215 

note  he  writes  the  words,  "on  demand"  in  place  of  "six 
months  after  date." 

It  is  sometimes  convenient  to  pay  a  note  in  parts,  or 
installment'.  Such  payments  are  called  Partial  Payments. 
They  should  be  written,  with  their  dates,  across  the  back 
of  the  note,  and  are  then  called  indorsements. 

Suppose  the  above  note  to  have  the  following  indorse- 
ments: 

Nov.  8,  1885,  received  $250. 

Apr.  14 ^  1886,  received  $150. 

Write  the  note  on  paper  and  put  on  the  indorsements. 

What  money  was  due  Nov.  8,  1885? 

What  was  due  after  the  payment  of  that  date? 

What  was  due  on  the  remainder  Apr.  14,  1886? 

What  was  still  due  after  the  payment  of  that  date  ? 

All  payments  must  first  go  towards  paying  interest  due. 
If  a  payment  is  not  enough  to  pay  the  interest,  it  is  counted 
with  the  next  payment,  and  its  date  left  out. 

Suggestion. — The  teacher  may  ask  the  trustees  to  purchase  a 
book  of  note  blanks  for  the  practical  use  of  classes.  Five  of  the 
following  notes  should  be  written  on  the  printed  blanks. 

EXERCISE   240.    (Written.) 

Write  out  the  following  in  proper  form  on  paper,  placing 
the  indorsements  on  the  back,  and  perform.  Determine 
mentally,  by  inspection,  whether  a  partial  payment  is  too 
small  to  be  taken  out  t"^  itself. 

1.  Date,  Jan.  1^  188:  Place,  your  own  town.  Face, 
$1500.  Interest,  6%.  :..iorsements:  Aug.  7,  1885,  $500. 
Dec.  7,  1885,  $500.     What  is  due  Jan.  1,  1886? 

2.  Face,  $480.  Mar.  8,  1884.  Interest,  1%.  Indorse- 
ments: •  Sept.  3,  1884,  $196.80.  Mar.  3,  1885,  $214.  Sept. 
3,  1885,  paid  the  amount  due.     Find  it. 

3.  Face,  $1000.     July  20,  1884.     Interest  at  8/"^.     In- 


216  CALIFORNIA   SERIES. 

dorsements:  Mar.  5,  1885,  $50.     July  5, 1885,  $450.     What 
was  still  due? 

4.  Face,  $1230.  Date,  Jan.  1,  1886.  Interest  at  b\%. 
Indorsements:  Mar.  1,  1886,  $98.  June  7,  1886,  $500. 
Sept.  20,  1886,  $290.  Dec.  10,  1886,  $100.  What  is  due 
Jan.  1,1887? 

5.  Face,  $800.  Date,  Mar.  1,  1886.  Interest  at  10%, 
Indorsements:  Aug.  10,  1886,  $200.  Sept.  1,  1886,  $50. 
Jan.  1,  1887,  $15.     What  was  due  Mar.  1,  1887? 

6.  Face,  $365.  Date,  July  10,  1885.  Interest  at  Q>%. 
Indorsements:  Sept.  10,  1885,  $68.65.  Nov.  18,  1885, 
$103.40.     What  was  still  due? 

7.  Face,  $2500.  Date,  Aug.  5,  1885.  Interest  at  1%. 
Indorsements:  Jan.  1,  1886,  $500.  March  10,  1886,  $750. 
Find  the  sum  due  Aug.  5,  1886. 

8.  Face,  $960.  Date,  June  25,  1886.  Interest  at  7^%. 
Indorsements:  Sept.  1,  1886,  $10.  Dec.  1,  1886,  $360. 
Jan.  1,  1887,  $300.     What  was  still  due? 

9.  Face,  $500.  Date,  Feb.  1,  1884.  Interest  at  8%. 
Indorsements:  Mar.  1,  1884,  $100.  Apr.  1,  1884,  $100. 
May  1,  1884,  $100.     What  was  due  June  1,  1884? 

10.  Face,  $1200.  Date,  May  15,  1886.  Interest,  6%. 
Indorsements:  Aug.  10,  1886,^  $500.  Nov.  1,  1886,  $500. 
What  was  due  Jan.  1,  1887? 

EXERCISE  241.    (Written.) 

Write  3  notes  of  your  own,  put  2  indorsements  on  each, 
perform,  and  bring  to  the  class  for  dictation. 


COMPOUND  INTEREST. 

Sometimes  when  a  note  specifies  that  interest  on  it  is  to 
be  paid  yearly,  semi-yearly,  quarterly,  or  the  like,  a  special 
agreement  is  made  that  if  such  interest  is  not  paid  when 


ARITHMETIC.  217 

due  it  shall  be  added  to  the  principal,  and  the  amount 
becomes  a  new  principal  for  the  next  period. 

This  method  of  computing  interest  is  called  Compound 
Interest.     In  many  states  it  is  prohibited  by  law. 

Compound  the  interest  at  8X  on  $540  for  7  mo.  12  da., 
payable  quarterly. 

OPERATION. 

$540  ^Principal. 
1.02  =am't  of  $1  for  i  year. 


$550.80=:"       "$540  for  i  year,  or  Prin.  for  2d  quarter. 


1.02  =  "       '•'  $1         ''    i 


561.82  =  "       "  $550.80  for  iyr.,  or  Prin.  for  3d  quarter. 

1.009i=  "       '•'  $1  for  1  mo.  12  da. 
56  7.06  =  "       "  $540  for  7  mo.  12  da.,  int.  computed 

quarterly. 

Find  the  compound  interest  above. 

EXERCISE  242.    (Written.) 
Find  the  compound  interest  on: 

1.  $1000.  for  4  yr.,  at  (S%,  payable  annually. 

2.  $300,  for  1  yr.  7  mo.,  at  8%,  payable  semi-annually. 

3.  $425,  for  11  mo.,  at  A%,  payable  quarterly. 

4.  $250,  from  Jan.  1,  1886,  to  Feb.  1,  1887,  at  b%,  pay- 
able semi-annually. 

5.  $500,  from  May  1,  1885,  to  Aug.  1,  1887,  at  Q>%,  pay- 
able annually. 

6.  $490,  for  8  mo.,  at  8%,  j^ayable  quarterly. 

7.  $1500,  from  Aug.   1,  1886,  to  Apr.  10,  1887,  at  7%, 
payable  semi-annually. 

8.  $275,  for  9  mo.,  payable  quarterly,  at  65^0. 

9.  $800,  for  2^  yr.,  payable  yearly,  at  Q>%. 

10.  $1200,  for  1  yr.  6  mo.  6  da.,  at  6>o,  payable  semi- 
yearly. 


218  CALIFORNIA    SERIES. 

Discounting  commercial  paper. 

Sacramento,  Cal.,  Mar.  4,  1885. 
$15003^0-. 

Six  months  after  date,  I  promise  to  pay  to  the  order  of 
James  Kenney,  Fifteen  Hundred  y\%  Dollars,  value  received. 

Allen  Paine. 

Suppose  James  Kenney  carries  the  above  note  to  the 
bank,  April  4,  to  get  money  on  it.  The  bank  will  deduct 
from  the  face  a  certain  per  cent,  say  1%  per  month,  from 
April  4  to  the  date  of  maturity,  and  pay  him  the  balance. 
Find  the  balance  on  this  note. 

This  is  called  discounting  the  note. 


Observe.  ^ 


1.  The  discount  is  made  on  the  face  of  non-inter- 

est bearing  notes. 

2.  Wheyi  the  note  bears  interest  the  discount  is 

made  on  the  amount  of  face  and  interest  at 
maturity. 


In  some  of  the  United  States  three  days,  called  days  of 
grace,  are  allowed  for  the  payment  of  the  note  after  it  is 
actually  due,  discount  being  made  for  the  extra  time. 
Days  of  grace  are  not  allowed  in  California. 

EXERCISE  243.    (Written.) 

Write  out  the  following  notes  on  paper  and  find  the  sum 
allowed  on  each  at  the  bank: 

1.  Note  of  $700,  Apr.  10,  1885,  payable  4  mo.  from  date. 
Discounted  at  8%,  June  10,  1885. 

2.  Note  of  $850,  July  3,  1885,  payable  60  days  from  date. 
Discounted,  Aug.  1,  at  1%'  a  month. 

3.  Note  of  $1400,  May  19,  1886,  bearing  interest  at  S%, 
payable  6  mo.  from  date.  Discounted,  Aug.  19,  1886, 
at  8^. 

4.  Note  of  $900,  June  1, 1885,  bearing  interest  at  1%  per 


ARITH^IETIC.  219 

month,  payable  3  months  from  ^is^e.     Discounted,  July  1, 
2ii\%  per  month.  "-— -:  -.^:_— --^ 

5.  Note  of  $250,  Sept  9,  1881,  payable  30  days  after  date. 
Discounted,  Sept.  9,  at  1%. 

6.  Note  of  $1850,  May  1,  1885,  payable  3  mo.  from  date. 
Discounted,  July  8,  at  1%  a  month. 

7.  Note  of  $525,  Jan.  5,  1886,  bearing  interest  at  \%  a 
month,  payable  4  mo.  from  date.  Discounted,  Feb.  5,  at 
1%  per  month. 

8.  Note  of  $300,  Dec.  11,  1886,  bearing  interest  at  |%  a 
month,  payable  in  6  mo.  Discounted,  Mar.  1,  1887,  at  \\% 
a  month. 

9.  Note  of  $1140,  Nov.  28,  1885,  bearing  interest  at  8%, 
payable  1  yr.  from  date.     Discounted,  Jan.  1,  at  %%. 

10.  Note  of  $1375,  Aug.  5,  1886,  payable  3  mo.  from 
date.     Discounted,  Sept.  1,  at  10%. 

11.  Note,  $735,  Jan.  13,  1886,  interest  at  10%,  payable  3 
months  from  date.     Discounted,  Feb.  25,  at  2%  per  month. 


DISCOUI^T. 

In  buying  a  bill  of  goods,  a  discount  or  discounts  are 
often  allowed  on  the  list  or  marked  price  of  the  goods,  and 
a  further  discount  on  the  result  for  cash.     Thus, 

Bought  a  bill  of  goods  amounting  to  $800  at  20  and  5  off, 
and  5%  off'  for  cash. 

FIRST    OPERATION. 

5)$ 800       ==marked  price  of  goods. 
160       =20%  discount. 
20)$640 

3  2        =b%  discount. 
20)$608 

ZQAO=b%  off  for  cash. 
$577.60  =actual  cost  of  the  goods. 


220  CALIFORNIA   SERIES. 

SECOND    OPERATION. 

2 

^00X4x19-^19      _^  ^^ 
5X^0X20 

/-vi  (Each  discount  is  reckoned  by  itself  and  on  the  sinn 

\     remaining  after  the  preceding  discount. 

EXERCISE    244.    (Written.) 

I.  Find  the  actual  cost  of  a  bill  of  goods  marked  at  $450 
at  40%  off,  and  b%  off  for  cash. 

^^  2.  Sold  a  bill  of  merchandise  at  2r)%  off,  and  5%  off  for 
cash;  find  the  whole  discount. 

3.  Sold  a  bill  of  goods  marked  at  $250  for  30,  and  5  off. 
Was  the  actual  selling  price  more  or  less  than  if  a  discount 
of  35%  had  been  made? 

4.  By  getting  a  discount  of  10,  and  10  off  for  cash,  I  pay 
$810  for  a  bill  of  goods;  what  was  the  list  price? 

5.  Bought  furniture  to  the  amount  of  $200,  on  which  a 
discount  of  5%  was  made  for  cash;  what  was  the  cost? 

6.  For  what  must  I  sell  goods  which  were  sold  me  for 
$830,  list  price,  at  30,  10,  and  5  off,  to  gain  20%  ? 

7.  Paid  $76  for  a  bill  of  glass  after  a  deduction  of  5%; 
what  was  the  invoice  price  ? 

8.  Find  the  cash  value  .of  a  bill  of  cloth  amounting  to 
$425.50  at  a  discount  of  10%,  and  5%  off  for  cash. 

9.  Bought  a  bill  of  goods  aniounting  to  $725  on  6  mo. 
credit,  on  which  a  discount  of  3%  was  allowed  for  cash; 
what  did  I  pa}^  for  the  goods? 

10.  The  retail  price  of  a  certain  book  is  $5.50.  If  I  get 
a  discount  of  10,  and  10  off  for  cash,  what  do  I  pay  for  the 
book  ? 

II.  I  paid  $1.50  for  a  book  after  a  discount  of  25%,  and 
16|%  off;  what  was  its  marked  price? 

12.  Sold  a  bill  of  goods  for  $700  on  G  mo.  at  15  off,  and 
deducted  4%  for  cash;  what  did  I  receive? 


ARITHMETIC. 


221 


ACCOUNTS. 

Every  one  who  receives  and  spends  money  should  keep 
a  record  of  receipts  and  expenses,  specifying  the  date  and 
nature  of  each  transaction. 

What  does  the  word  "cash"  mean?  Are  greenbacks 
cash?     Bank  checks?     Postage  stamps? 

The  following  is  a  record  of  a  boy's  receipts  and  expenses: 

Jan.  1,  1886,  money  on  hand,  $2.65.  Jan.  2,  paid  5c.  for 
marbles  and  10c.  for  lead  pencil.  Jan.  4,  paid  25c.  for  a 
Speller.  Jan.  5,  paid  10c.  for  a  bottle  of  ink.  Jan.  6, 
received  25c.  for  blacking  father's  shoes  one  week.  Jan.  7, 
paid  10c.  for  a  top  and  15c.  for  marbles.  Jan.  9,  received 
$1  for  driving  cow  to  pasture  and  50c.  for  milking.     Jan. 

11,  paid  40c.  for  a  Reader  and  60c.  for  an  Arithmetic.     Jan. 

12,  sold  top  for  5c.  Jan.  13,  paid  10c.  for  postage  stamps. 
Jan.  14,  paid  20c.  for  candy.  Jan.  16,  received  10c.  for 
doing  errands  and  paid  5c.  for  marbles.  Jan.  19,  lost  10c. 
Jan.  20,  received  40c.  for  blacking  father's  shoes.  Jan.  21, 
paid  50c.  for  a  kite.  Jan.  25,  sold  5  cents'  worth  of  mar- 
bles. Jan.  26,  received  25c.  for  clearing  the  yard.  Jan. 
27,  paid  15c.  for  setting  a  broken  light  of  glass.  Jan.  29, 
found  25c.     Jan.  30,  paid  $1.15  for  a  Geography. 

Obtain  paper,  rule  as  below,  copy,  and  fill  out  the  month's 
items.  Find  out  how  much  more  he  received  than  paid, 
see  if  it  agrees  with  the  balance,  then  add  each  column 
and  place  the  result  below. 


1886. 


CASH. 


Rec'd.        Paid. 


Jan. 

1 

(( 

o 

<( 

4 

On  hand, 

Marbles,  5  cents;    Lead  pencils,  Ij  cents. 
Speller, 

Carried  forward. 


$ 

2 

11   2, 

ct. 
65 

65 

$ 

ct. 


15 


zo 


40 


222 


CALIFOnNIA   SERIES. 


Jan, 

5 

a 

6 

a 

7 

i  i 

9 

I  i 

11 

Jan. 

30 

Brought  forward, 
Bottle  ink, . 

Blacking  father's  shoes, 

Top,  10  cents;  Marbles,  15  cents,      .     . 

Driving  cow,  $1 ;  Milking,  50  cents,      .     . 
Reader,  40  cents;  Arithmetic,  60  cents. 


Balance, 


$ 

ct. 

$ 

2 

65 
25 

1 

50 

1 
1 

5 

50 

5 

ct. 
40 
10 

25 

00 


55 

50 


What  does  the  "balance"  in  the  above  account  show? 
If  the  amount  of  money  on  hand  does  not  agree  with  bal- 
ance, what  does  the  difference  show  ?  In  the  account,  which 
column  is  the  larger?  Could  the  other  column  ever  be 
larger  in  a  "cash"  account?     Whyf 

The  "balance"  should  be  found  twice  a  month,  at  least, 
and  oftener  as  the  business  is  larger.  Business  firms  and 
banks  balance  their  "  cash  "  every  day.  It  is  well  to  write 
the  balance  in  red  ink.     Why? 

Open  an  account  for  February  wdth  the  above  balance  on 
hand,  and  write  items  of  your  own.  Take  care  that  at  no 
time  your  "  paid  "  items  exceed  the  "  received  "  items.  Bal- 
ance and  bring  to  the  class. 

Write  out  the  following  "cash"  acct.  of  a  teacher,  and 
balance  every  Saturday: 

May  1  (Sat.)  1886,  Cash  on  hand,  $78.80.  May  3,  Bought 
20  cents  worth  of  P.  O.  stamps.  ^lay  4,  Paid  $5  borrowed 
money.  May  5,  Bought  11  yards  cashmere  @  $1.25;  pair 
of  shoes  $4.50;  1  doz.  hdkfs.  $1.75.  May  6,  Paid  express 
on  package  of  books  25  cents.  May  7,  Sent  by  money  order 
$2.75  to  pay  for  books,  paying  10  cents  for  the  order.  May 
8  (Sat.),  Paid  for  postal  cards  10  cents;  stamped  envelopes 
55  cents;  note  paper  60  cents. 


ARITHMETIC.  223 

May  11,  Paid  2  weeks'  board,  to  May  15,  @  $4.50.  May 
13,  Paid  spool  thread  10  cents;  bottle  mucilage  25  cents. 
May  15  (Sat.),  Carriage  hire  $2.50;  received  $9.25  for  serv- 
ices on  Board  of  Education. 

May  18,  Paid  2  weeks'  board  to  May  29.  May  19,  Paid 
mo.  contribution  to  church  $1.50;  gave  a  poor  woman  50 
cents.  May  20,  Paid  for  sending  telegram  75  cents;  crack- 
ers 25  cents.  May  22,  Paid  2  mo.  subscription  to  "  Daily 
Herald"  @  65  cents;  received  mo.  salary  $75. 

May  24,  Deposited  in  bank  $50.  May  26,  Paid  $1  for 
book,  25  cents  for  "  legal  cap,"  40  cents  for  ribbon.  May 
27,  Paid  $1.25  for  gloves;  exchanged  a  second-hand  Reader 
for  a  new  one  worth  60  cents,  being  allowed  25  cents  for  the 
old  one,  and  paid  the  difference.  May  28,  Lent  a  friend 
$2.  May  29,  Paid  for  pins  10  cents,  penknife  50  cents, 
sheet  music  30  cents. 

Balance  shows  my  pocket-book  5  cents  short,  for  which  I 
can  not  account.     Balance. 

Write  out  the  following  account: 

July  1,  1887,  received  $5.  July  4,  bought  5  flags  at  25c. 
each,  3  bunches  of  fireworks  at  30c.  a  bunch,  18  yd.  bunt- 
ing at  10c.  per  yd.  July  6,  earned  20c.  selling  papers. 
July  7,  gave  a  poor  woman  10c.     Balance. 

Write  out  a  cash  acct.  of  your  own.  Begin  with  $5  on 
hand.     Have  6  items  received,  and  8  paid.     Balance. 

What  is  a  debt?  A  debtor?  A  credit?  A  creditor?  Why 
is  it  necessary  to  keep  an  account  of  our  debts  and  credits? 
When  is  a  man  your  debtor?  Your  creditor?  Is  John 
Smith  debtor,  or  creditor,  for  what  we  give  him  ?  For  what 
he  gives  us?  What,  then,  does  the  debtor  side  of  a  man's 
account  show  ?  The  creditor  side  ?  If  the  debtor  side  be 
the  larger,  what  does  the  balance  show?  If  the  creditor 
side  be  the  larger?  If  both  sides  are  equal?  Explain  each 
item  in  the  following  account,  which  we  will  suppose  to  be 
your  account  with  John  Smith: 


224 


CALIFORNIA    SERIES. 


Dr. 


JOHN   SMITH. 


Cr. 


188G. 
Jan. 


He  owes  me  . 
2  loads  hay  . 
Use  of  wagon 
Cash   .... 


$ 

ct. 

188G. 

43 

05 

Jan. 

2 

9 

50 

u 

12 

50 

a 

18 

9 

70 

i( 

27 

t( 

31 

~~ 

Cash 

Work  with  team 

Calf 

Order  on  Robert 
Stewart   .     .    . 

Balance  .... 


28 


ct. 
GO 
75 
00 

90 


Copy  the  above  account  and  complete  it.  Change  it  so 
as  to  show  John  Smith's  account  with  you.  Write  an  im- 
aginary continuation  of  the  account  during  the  month  of 
February.  Have  5  Dr.  items  and  5  Cr.  items,  and  have  .1^5 
due  John  Smith  Mar.  1.     Have  no  dates  on  Sunday. 


Observe. 


i 


Any  person  becomes  Dr.  for  goods  or  money  deliv- 
ered TO  him. 
Any  person  becomes  Cr.  for  goods  or  money  deliv- 
[^      ered  by  him. 


The  following  are  the  transactions  of  a  farmer  with  a 
merchant,  S.  C.  Griggs  &  Co.  Copy  as  above,  writing  the 
account  for  each  party: 

1886.  Mar.  1,  Sold  S.  C.  Griggs  &  Co.  7  doz.  eggs  @  18 
cents;  11  rolls  butter  at  40  cents.  Received  10  lb.  sugar  @ 
8  cents;  1  sack  salt  25  cents.  3.  Delivered  them  10 
sacks  potatoes  @  85  cents.  4.  Bought  20  yd.  sheeting  @ 
12^  cents;  12  yd.  print  @  10  cents.  6.  Sold  12  doz.  eggs 
@  16  cents;  9  rolls  butter  @  40  cents;  10  sacks  potatoes  @ 
80  cents.  Bought  4  50-ib.  sacks  flour  @  $1.12^;  2  lb.  tea 
@  65  cents;  5  lb.  coffee  @  37-|  cents.  9.  Bought  1  box 
soap  $1.15;  2  lb.  cheese  @  17^  cents.  12.  Sold  10  doz. 
eggs  @  18  cents;  13  rolls  butter  @  40  cents;  5  sacks  pota- 
toes @  90  cents.  15.  Bought  20  lb.  dried  apples  @  7 
cents;  can  lard  65  cents;  2  boxes  paper  collars  @  15  cents; 
5  cans  apricots  @  30  cents.     18.     Sold  2  loads  wood  @ 


ARITHMETIC.  225 

$4.50.  22.  Bought  1  lamp  $2;  1  pr.  boots  $5.50;  1  ham 
12  lb.  @  18  cents.  25.  Sold  15  doz.  eggs  @  20  cents;  8 
rolls  butter  @  50  cents.  'Bought  suit  clothes  $8;  8  lb. 
sugar  75  cents;  1  10-gallon  can  kerosene  $1.75;  1  pr.  boys' 
shoes  $3.50;  1  sack  oatmeal  50  cents.     Balance. 

Write  an  imaginary  account  between  the  nearest  mer- 
chant and  yourself.  Have  8  purchases  and  7  sales.  Have 
your  prices  reasonable  and  the  transactions  such  as  you 
might  make. 

Sometimes  a  person  engages  in  an  enterprise,  like  renting 
or  purchasing  grain  land,  on  which  he  wishes  to  know  his 
profit  or  loss  over  and  above  interest  on  the  money  invested. 

The  following  is  an  account  of  the  expenses  and  returns 
of  a  barley  field.  Use  the  name  "  Barley  field,"  debit  it 
with  all  its  expenses,  including  the  interest  on  the  value 
of  the  land  @  6X  for  a  year,  and  credit  it  with  all  its 
returns. 

Balance,  and  find  the  per  cent  of  profit  on  the  land  value. 

160  acres  of  land  valued  at  $70  per  acre.  Plowing,  $1.30 
per  A.;  sowing,  10  cents  per  A.;  seed,  $1  per  A.;  harrow- 
ing, 25  cents  per  A.;  poisoning  squirrels,  $4.50;  heading, 
$1.75  per  A.;  thrashing,  10  cents  per  cental,  2700  centals; 
sacks,  8  cents  each,  averaging  135  lb.  to  a  sack;  sack  twine, 
$8;  hauling  grain  to  warehouse,  5  cents  a  sack;  sold  the 
lot  at  the  warehouse  at  $1.01-|  per  cental;  sold  the  straw 
and  stubble  for  $95. 

Do  the  same  with  the  following  Dairy  account: 

40  cows  at  $35  per  head. 

1886.     Jan.  1.    Salt,  $1.     5.   Rennets,  $1.30;  coloring,  50 

cents.     11.     Wood,  $5.     12.     Cheese  bandages,  $7.20.     18. 

Sold  1600  ft),  cheese  @  9  cents;  freight  and  commission, 

1  cent  per  ft).     30.     Paid  2  men's  wages,  $50;  board,  $32; 

pasture  for  Jan.,  $1  per  head.     Feb.  1.  Sold  1400  lb.  cheese 

@  9^  cents;  freight  and  com.,  1  cent  per  ft).     Balance. 
15— A 


226 


CALIFORNIA   SFEIES. 


BALANCE    SHEET. 


The  following  "Balance  Sheet"  is  a  statement  of  Luke 
Smith's  debts  and  credits  at  the  beginning  of  the  year. 
Copy  on  the  board  and  explain  each  item: 


1886. 

BALANCE  SHEET. 

Debts. 

Credits. 

1 

$ 

ct. 

$ 

ct. 

Jan. 

1 

Farm  and  improvements, 

758 

75 

Household  proj^erty,     .     . 

176 

50 

Mortgage  on  farm,      .... 

425 

85 

Note  payable  on  demand,     .     . 

56 

50 

John  Mason,      .... 

43 

65 

Wm.  Jones, 

88 

35 

Chas.  Bell's  note,  .     .     . 

76 

50 

Cash, 

19 

85 

Bank  of  California,     .     . 

78 

95 

Balance, 

Put  into  a  balance  sheet  the  following  statement  of  Luke 
Smith's  debts  and  credits  Feb.  1,  1886: 

Farm,  $472.  Improvements,  $326.75.  Household  prop- 
erty, $176.50.  Mortgage,  $395.25.  John  Mason  owes  him 
$29.70.  He  owes  Chas.  Bell  $18.25  and  Thomas  Olmstead 
$29.85.  He  has  $28  in  money  and  $48.25  in  the  Bank  of 
California. 

Compare  this  with  the  preceding  month  and  tell  whether 
he  has  gained  or  lost.  How  does  he  stand  with  each  per- 
son Feb.  1,  as  compared  with  his  standing  Jan.  1?  If  his 
debts  were  larger  than  his  credits,  how  would  he  settle  with 
his  creditors? 

Write  an  account  on  balance  sheet  of  your  own  for  Luke 
Smith  for  March.     Leave  him  in  debt. 

What  is  a  bank?  What  use  have  we  for  it?  How  does 
the  bank  get  pay  for  taking  care  of  our  money? 

If  you  wish  to  pay  a  person  a  debt  and  have  money  in 


ARITHMETIC. 


227 


the  bank,  instead  of  paying  liim  in  money  you  can  write 
an  order  on  your  banker  to  pay  the  same. 

Such  an  order  upon  a  bank  is  called  a  check. 

When  you  deposit  money  in  the  bank,  the  bank  gives 
you  a  written  statement  to  that  effect,  called  a  certificate 
of  deposit,  and  you  draw  the  money  on  presenting  this 
certificate. 

Or  the  bank  will  give  you  a  bank  account  book,  and  you 
may  draw  checks  till  the  money  is  all  drawn  out. 

[Form  of  check.] 

No.  9.  Merced,  Cal.,  May  7,  1886. 

FIRST  NATIONAL  BANK 

Pay  to  James  Cash  or  Order,  One  Hundred  Thirteen  and-f-^^ 
Dollars. 

$113.50.  John  Simms. 

Copy  the  following  bank  account  and  explain  each  item. 
Write  out  the  checks  on  paper,  with  yourself  as  depositor. 
Add  10  items  and  balance  with  $75  to  your  credit  in  bank. 


1886. 


BANK  OF  VENTURA. 


Br. 


Cr. 


Jan. 


Gold    ^ 

Check    I  Tlios.  Cruson  .... 

Check  II  Wm.  Bell  &  Co.  .  .  . 
Silver 

Check  III  Bartlett  Bros.,    .     .     . 

Check  IV  Self 

Check  on  Bank  Cal,, 

M.  Wooley's  check  on  Bank  Vent., 


100 


46 


14 

9 


ct. 

00 

50 


ct. 

85 
05 

75 

65 


228 


CALIFORNIA   SERIES. 


EXCHANGE. 

Suppose  you  owe  A.  B.  Stanton  of  New  York  $500.  To  avoid 
inconvenience  and  risk  of  sending  the  money  you  may  buy 
of  your  banker,  say  D.  B.  Fairbanks,  an  order  on  some  New 
York  banker,  say  S.  A.  Spring,  to  pay  A.  B.  Stanton. 


ARITHMETIC.  220 

You  send  the  order  to  A.  B.  Stanton:  and  he,  on  receiv- 
ing it,  presents  it  to  S.  A.  Spring.  Spring  writes  acceptance 
across  the  face  as  above,  if  willing  to  pay  it.  At  maturity, 
30  days  from  acceptance,  Stanton  presents  the  order  and 
receives  the  money.  If  he  wishes  the  money  before  ma- 
turity, the  banker  w^ill  discount  it  for  the  difference  in 
time. 

Such  an  order  is  called  a  draft,  or  bill  of  exchange  ;  and 
this  method  of  making  payments.  Exchange. 

A  draft  is  always  made  out  in  the  money  of  the  country 
on  which  it  is  drawn. 

Drafts  are  either  "  sight"  or  "  time"  drafts;  that  is,  pay- 
able on  presentation,  or  at  a  certain  specified  time  after 
presentation. 

Which  is  the  above  draft? 

The  maker  of  a  draft  is  called  the  drawer;  the  person 
to  whom  addressed,  the  drawee ;  and  the  person  to  whom 
payable,  the  payee. 

Name  each  in  the  above  draft. 

A  draft  may  be  transferred,  like  a  note,  by  indorsement. 

If  the  merchants  of  New  York  owe  the  merchants  of  San 
Francisco  more  than  San  Francisco  merchants  owe  them, 
bills  of  exchange  on  New  York  will  be  plentiful  in  San 
Francisco  and  can  be  purchased  cheaply,  or  at  a  discount ; 
if  the  balance  is  due  the  other  way,  bills  of  exchange  on 
New  York  will  be  scarce  in  San  Francisco,  and  will,  there- 
fore, be  dear,  or  at  a  'premium. 

Time  drafts  are  discounted  to  the  buyer  for  the  time 
specified.  The  time  discount  is  understood  to  be  the  rate 
for  1  year,  unless  otherwise  stated. 

All  discounts  or  premiums  are  reckoned  as  per  cent  of 
the  face  of  the  draft.  The  above  draft  on  S.  A.  Spring  has 
a  time  discount  at  7%;  if  it  be  purchased  at  1%  premium 
what  is  paid  for  it?     At  1%  discount? 


230  CALIFORNIA    SERIES. 

EXERCISE  245.    (Written.) 

Write  out  the  following  drafts  to  imaginary  payees  and 
drawers,  with  proper  acceptance  in  red  ink.  Write  no  accept- 
ance on  sight  drafts.     Why  ? 

1.  Find  the  cost  in  New  Orleans  of  a  draft  for  $5000  on 
New  York  at  60  days'  sight,  exchange  being  1^%  premium, 
interest  at  S%  per  annum. 

2.  Bought  a  sight  draft  on  St.  Louis,  for  $580,  at  ^%  dis- 
count; what  was  the  cost? 

3.  I  paid  $2481.25  for  a  sight  draft  on  Chicago,  at  i% 
discount;  what  was  the  face  of  the  draft? 

4.  I  wish  to  buy  a  60  days  draft  on  London,  for  £320, 
exchange  at  $4.95  per  £,  interest  at  7%;  what  will  it  cost? 

5.  Paid  $1566.15  for  a  sight  draft  on  Boston,  at  1^%  dis- 
count; what  was  the  face? 

6.  Paid  $4500  for  a  draft  on  New  York  at  90  days  sight, 
premium  1^%,  interest  at  Q%  per  yr.;  find  the  face. 

7.  Find  the  cost  of  a  sight  draft  on  Paris  for  4000  francs 
at  1%  discount. 

8.  A  sight  draft  for  $800  cost  me  $794;  what  was  the  rate 
of  discount? 

9.  What  is  the  cost  of  a  10  days  sight  draft  for  $765,  at 
^%  premium,  time  discount  8^^-^? 

10.  The  cost  of  a  30  days  draft  for  $800,  time  discount 
including  grace  6%,  was  $799.60;  what  was  the  rate  of  dis- 
count or  premium  ? 

11.  I  buy  in  Sacramento  a  45  days  draft  on  Paris  for 
1000  francs,  interest  1%  a  month,  exchange  1^%  premium; 
what  do  I  pay? 

12.  I  pay  $162.75  for  a  draft  on  Paris  at  45  days  after 
date,  time  discount  1%  a  month,  exchange  l-h%  premium; 
what  is  the  face  of  the  draft? 

13.  I  buy  in  Paris  a  60  days  draft  on  T^ondon  for  £500, 
exchange  being  at  26  francs  per  £,  time  discount  5%,  what 
do  I  pay?     Is  exchange  at  a  discount  or  premium? 


ARITHMETIC.  231 

The  payment  of  small  sums  at  a  distance  is  often  effected 
by  means  of  postal  money  orders  or  by  bank  checks. 

A  money  order  is,  in  effect,  a  sight  draft  drawn  by  the 
postmaster  of  the  debtor  upon  the  postmaster  of  the  cred- 
itor; payable  to  the  creditor,  or  order. 

Name  the  payer ^  drawer,  and  payee. 

Money  orders  are  subject  to  the  following  charges  and 
regulations : 

On  orders  not  exceeding  $10 8  cents. 

Over  $10  and  not  exceeding  $15 10  " 

"       15    "       "          "            30 15  " 

30    "       "          "            40 20  " 

"       40    "       "          "             50 25  " 

50    "       "          "            60 30  '' 

"       60    "       "          "            70 35  " 

70    "       "          "            80 40  " 

80    "       "          ''          100 45  '' 

A  single  order  may  include  any  amount,  to  $100. 

Not  more  than  3  orders  may  be  issued  in  one  day,  to  the 
same  applicant,  payable  at  the  same  office,  to  the  same 
payee. 

A  money  order  is  negotiable,  but  subject  to  one  transfer 
only. 

A  check  is,  in  effect,  a  sight  draft  on  a  bank. 

The  value  of  a  check  as  a  medium  of  exchange  is,  that 
it  passes  for  money,  when  certified  or  signed  by  the  cashier 
of  the  bank  on  which  it  is  drawn,  and  properly  indorsed. 

Such  a  check  is  called  a  certified  check,  and  is  usually 
cashed  by  any  bank  at  which  it  is  presented,  without  dis- 
count to  one  who  keeps  an  account  with  that  bank.  To  one 
not  keeping  an  account  with  that  bank,  it  is  customary  to 
discount  it  at  20  ct.  or  25  ct.  per  $100,  and  a  like  rate  is 
charged  the  buyer  of  such  a  check  by  a  bank  with  which 
he  does  not  keep  an  account. 


232 


CALIFORNIA   SERIES. 


EXERCISE  246.  (Oral.) 

Name  the  charges  on  money  orders  for  the  following 
sums,  and  specify  if  it  takes  more  than  one  order  for  the 
amount  named : 

$219.00                $.25  $160.00 

175.00            200.00  80.50 

8.50            190.00  40.05 

3.50              60.00  100.10 


$2.50 

$20.00 

25.00 

40.00 

250.00 

125.00 

19.90 

140.00 

EXERCISE   247. 

Write  sight  drafts  for  the  following  sums  and  compute 
their  cost  at  i%  premium: 

$150.00         $375.00         $400.50            $110.00  $230.75 

190.00            75.25             20.00              318.00  500.00 

Write  a  draft  for  $325  at  15  days  sight,  time  discount 
12%,  exchange  i%  premium.     Compute  cost. 

AVrite  a  draft  for  $1000,  at  10  days  sight,  exchange  i% 
discount,  time  discount  9%.     Compute  cost. 

Write  a  draft  for  $725,  at  75  days  sight,  time  discount 
10%,  exchange  i%  premium.     Compute  cost. 


J 


ARITHMETIC.  233 


AVERAGE  OF  PAYMENTS. 

I  buy  2  bills  of  goods  Jan.  1  of  Mr.  A;  one  of  ^800  on  3 
nio.,  and  tbe  other  of  $250  on  4  mo.  If  I  pay  them  before 
they  are  due,  I  lose  the  use  of  the  money  for  the  remainder 
of  the  time.  If  I  delay  paying  them  after  they  are  due, 
Mr.  A  loses  the  use  of  the  money  for  the  time.  Now,  I  wish 
to  pay  both  debts  together,  without  loss  to  either  party. 

FULL   ANALYSIS. 

The  use  of  $300,  3  mo.=use  of  $1  900  mo.  (300  X  3  mo.) 
4  mo.=  "  "  $1  1000  mo.  (250  X  4  mo.) 


"   "  "  $550  j  ^  ^^^-  ^  =  The  use  of  $1  1900  mo.  =  use 
j  4  mo.  \ 

of  $550  5^^  of  1900  mo.  =  3fj-  mo.  3yV  mo.=:  3  mo.  14 

da.,  +  Jan.  1  =  Apr.  15. 

CONTRACTED  OPERATION. 

3x300=  900  mo. 
4X250  =  1000  mo. 

550  )1900mo. 

3^^  mo.  ■=  3  mo.  14  da.     Average  Time. 

Jan.  1+3  mo.  14  da.  =  Apr.  15,  Date  of  Payment. 

EXERCISE    248.    (Written.) 

1.  I  owe  $180  in  5  mo.,  $250  in  8  mo.,  and  $100  in  9  mo. 
At  what  date  may  I  pay  the  whole  with  no  loss  ? 

2.  A  man  owes  a  note  of  $800  payable  in  3  mo.,  and  one 
of  $1000  payable  in  4  mo.  Find  the  average  time  of  pay- 
ment. 

3.  Bought,  Apr.  8,  of  C.  W.  Spring  &  Co.,  the  following 
bills  of  goods:  $150  on  3  mo.  credit;  $175  on  4  mo.  credit; 
and  $200  on  6  mo.  credit.  Find  the  average  time  and  date 
of  payment  for  all. 

4.  I  owe  2  bills  to  the  same  man,  one  of  $390  due  in  16 


234  CALIFORNIA    SERIES. 

days,  and  one  of  $475  due  in  20  days.     In  how  many  days 
may  I  pay  both  together? 

5.  Find  the  average  date  for  paying  3  bills  due  as  fol- 
lows: May  31,  $100;  June  18,  $150;  July  9,  $200.  (Com- 
pute each  from  May  31.) 

6.  If  I  borrow  $250  for  8  mo.,  how  long  should  I  lend 
$400  to  repay  me  an  equal  interest? 

7.  If  you  lend  a  friend  $550  for  6  mo.,  what  sum  should 
he  lend  you  for  10  mo.,  to  repay  the  favor? 

8.  A  man  owes  a  debt  of  $1000  on  10  mo.,  of  which  he 
pays  i  in  4  mo.  and  ^  in  8  mo.  When  is  the  remainder 
due? 

9.  Carry  out  the  items  in  the  following  bill  and  find 

when  it  is  due: 

San  Francisco,  Mar.  22,  1886. 
F.  E.  Adams  (Hollister), 

Bought  of  Ellis,  Wells  &  Co. 

100  yd.  broadcloth @    $4  on  2  mo. 

500    ''      sheeting @  16c.   "  3     '' 

75  pieces  fancy  goods @    $3    "  4     " 

10.  In  bill  3,  page  120,  assume  the  purchases  to  be  on  3 
months  time,  and  find  the  average  time  for  payment.  • 


ARITHMETIC.  235 


AVERAGE. 

Suppose  I  mix  together  2  lb.  of  tea  worth  60  cents  a  ib., 
4  ft),  worth  70  cents  per  ft).,  and  4  ft),  worth  80  cents  per  ft). 
What  is  the  w^eight  of  the  mixture?  Its  value?  Its  aver- 
age value  per  ib.? 

EXERCISE  249. 

1.  Sold  2  sheep  at  $2.50  per  head,  3  at  $3  per  head,  and 
10  at  $3.25  per  head.  What  was  the  average  price  per 
head? 

2.  Mixed  10  centals  of  wheat  worth  90  cents  per  cental,  8 
centals  worth  95  cents,  and  7  centals  worth  $1.  What  was 
the  value  of  the  mixture  per  cental? 

3.  Mixed  45  ft),  of  sugar  at  8  cents  per  lb.,  and  30  ib.  at 
10|  cents  per  ib.  For  what  must  I  sell  the  mixture  per  ib. 
to  gain  10%  ? 

4.  A  grocer  sold  8  rolls  of  butter  which  cost  him  40  cents 
per  roll,  and  10  rolls  that  cost  him  50  cents  per  roll,  all  at 
50  cents  per  roll.     What  was  his  average  gain  per  roll? 

5.  A  liquor  dealer  mixed  50  gal.  of  liquor  worth  35^cents 
per  gal.,  50  gal.  worth  42  cents  per  gal.,  50  gal.  worth  40 
cents  per  gal.,  and  50  gal.  of  water.  What  was  the  average 
value  per  gal.  of  the  mixture? 

6.  A  grocer  mixed  12  lb.  of  sugar  worth  6  cents  per  ib.,  9 
ib.  worth  8  cents  per  ib.,  15  ib.  worth  11  cents  per  ib.,  and  17 
lb.  worth  13  cents  per  ib.  What  w^as  the  value  of  the  mix- 
ture per  ib.  ? 

7.  A  confectioner  mixed  5  ib.  of  candy  at  40  cents  per  ib., 
7  ft),  at  25  cents  per  lb.,  10  ib.  at  20  cents  per  ib.,  and  2  ib. 
at  50  cents  per  ib.,  selling  the  mixture  at  30  cents  per  ib. 
Did  he  gain  or  lose,  and  how  much? 

Mix  4  kinds  of  sugar,  worth  respectively  7,  8,  12,  and  13 
cents,  so  that  the  mixture  shall  be  worth  11  cents  per  ib. 


236  CALIFORNIA   SERIES. 


WORK.  PROOF. 

Lb.         Gain  or  loss.  ^^'- 

7..1--.-+4  1®    7=   7 


-4-  /    Total  gain. 

13__3 —6 


1  "   12 -=12 
3^  "   13  ==3  9 

6  )  6  6  (   1   1    Ct.  Av.  price. 


— 7  Total  loss.       Explanation. — Taking  1  ft),  at  7  ct., 
the  gain  is  4  ct.;  and  1  lb.  at  8  ct.,  the  gain  is  3  ct.    Total  gain,  7  ct. 
Taking  3  lb.  at  13  ct.,  the  loss  is  6  ct.,  and  1  ft.  at  12  ct.  makes 
the  total  loss  7  ct. 

The  mixer  gains  on  all  goods  below  the  average  price, 
and  loses  on  all  above.  Any  set  of  numbers  which  makes 
his  gains  and  losses  equal,  is  correct.  Usually,  several  cor- 
rect sets  of  answers  may  be  found. 

Find  two  more  sets  of  correct  answers  to  the  above  exam- 
ple.    Test  each  set  by  the  proof  given  above. 

A  little  skill  will  always  enable  the  student  to  balance 
the  gains  and  losses,  using  whole  numbers.  If  this  be 
found  difficult,  make  the  last  number  of  pounds  fractional 
and  multiply  the  number  of  pounds  of  each  kind  by  the 
denominator  of  the  fraction. 

EXERCISE     250.     (Answers  Yariable.) 

1.  Mix  three  kinds  of  tea,  worth  55,  60,  and  70  cents,  to 
make  a  mixture  worth  65  cents. 

2.  If  3  ft),  of  the  55-cent  kind  is  used,  how  much  of  each 
of  the  others  must  be  used  ? 

3.  How  much  water  must  be  mixed  with  a  cask  of  wine 
containing  30  gal.  at  $1.50,  to  reduce  the  price  to  $1? 

Sometimes  one  or  more  of  the  quantities  may  be  limited. 

4.  Claret  worth  35  ct.,  40  ct.,  50  ct.,  and  56  ct.  per  gallon, 
is  to  be  mixed  with  20  gallons  @  64  ct.,  and  14  gallons  at 
70  ct.,  to  make  the  mixture  worth  52  ct.  per  gallon.  How 
many  gallons  of  each  shall  be  taken? 


ARITHMETIC.  237 


POWERS  AND  ROOTS. 

What  is  the  area  of  a  square  whose  side  is  8  inches? 

The  product  of  a  number  by  itself  is  called  the  square 
of  that  number.     Thus,  6^  is  the  square  of  8. 

The  number  itself  is  the  square  root  of  the  product. 
Thus,  8  is  the  square  root  of  64. 

The  square  is  indicated  thus:  8^^  9^  2S^. 

The  square  root  thus:  1/64,  y^81,  \/625;  or  64'-'^  81^,  6SS'''\ 

EXERCISE  251.    (Oral.) 

Name  the  results  indicated  by  the  signs  affixed  to  the 
following  numbers: 


49'^ 

r-' 

10^ 

8100^ 

i^y           .3^ 

? 

IH 

20'^ 

70-^ 

iiY'       .09^ 

v/25 

1/9 

1/100 

90'^ 

(1)^      •     .25^ 

5^ 

42 

30'^ 

1/3600 

i-hY'         -36^ 

64^ 

2' 

|/2500 

V400 

iW         1.1^ 

12^ 

1/I6 

1/I6OO 

4900^ 

ay     1.21^ 

9^ 

IV 

50^ 

100^ 

.1-           ^1.2^ 

121 

1/8I 

900^ 

60^ 

0.1^          1.44^ 

Refer  to  method  for  squaring  numbers  of  two  figures  on 

K  117, 

and  square 

i  the  numbers  from  14  to  19  inclusive  by 

hat  method. 

SQUARE  BOOT. 

To  extract  the  square  root  of  a  number. 

The  full  explanation  of  the  extraction  of  roots  must  be 
left  to  Algebra.  We  here  give  such  illustrations  as  will 
serve  to  fix  the  method  in  the  memory  and  give  a  practical 
explanation  of  it. 


238  CALIFORNIA    SERIES. 

Find  the  square  root  of  1024. 

FULL    OPERATION. 

Explanation. — 1024  =  tens2  +  2  x 
1  0  2 4. ( 30+2.    tens  X  units  +  units'^      The    largest 
900  tens'-'  in  1024  is  900  =  302.     The  re- 

2X  30  =  (30_)1^2T         niainder  124  =  2 xtens       contracted. 

]^20         (2  X  30  =  60)  X  units  +  10'^  4    (32 

T         units^.    Since  124  con-  „        '   — ~ 

tains  60  x  units,  units     — ■ 

^~=±_         =  124 --60  =  2  units,      §A)  ^  ^^ 
0  with  a  remainder  4,  12  4 

which  is  units^. 

The  O's  may  be  omitted  in  the  operation,  and  because  60 
and  2  are  each  multiplied  by  2,  both  may  be  multiplied  at 
once,  as  shown  in  the  contracted  work.  In  dividing  by  6, 
remember  it  is  6  tens  in  12  tens  and  not  6  in  124.  Omit 
mentally  the  right  hand  dividend  figure. 

This  operation  may  be  extended  to  any  number. 

In  squaring  a  number,  as  48.6, 

.6^^=  .36         We  see  that  the  square  of  each  figure 

8'^ .    =      64  .  occupies  two  places.     Hence  point 

4^      .    =16       .  off  the  number,  whose  square  root 

is  to  be  found,  into  groups  of  2  figures  each,  commencing  at 
the  decimal  point.  Make  full  groups  at  the  right  of  the 
point  by  annexing  a  0  if  necessary.  You  will  notice  that  in 
finding  each  figure  of  the  root,  you  use  the  group  containing 
its  square. 

Find  the  square  root  of  2361.96. 
operation. 

23'61.'96(48^ 

^6 

88.)  761. 
704. 


I 


96.6)     57.96 
57.96 


ARITHMETIC. 


239 


1.  2401.^ 

2.  1.8225^ 

3.  930.25^ 

4.  .1296^ 

5.  1.225^ 

6.  7056/^ 

7.  .8201^^ 

8.  384736.^ 

9.  349281.^ 

10.  .4096^ 

11.  4.096^ 

12.  11881/^ 

2^,  8^,  12-^ 


EXERCISE  252.    (Written.) 

13.  1/17^  25. 

14.  i/1040yV  26. 

15.  1/424.36  27. 

16.  1/1.0675  28. 

17.  1/10575.  29. 

18.  |/.00625  30. 

19.  ^.0625  31. 

20.  1/46656.  32. 

21.  V1232136.  33. 

22.  1/163.84  34. 

23.  1/6.5536  35. 

24.  (Mfl)''  36. 
18^%  and  80'^  to  2  decimal  places. 


\9801/ 

1866.24^ 

9312.25^ 

315844.^ 

3858.^^ 

226576.^ 

28134.^^ 

.120409^ 

42.025^ 

4.2025^ 

516961.^ 

51696.  r^ 


PRACTICAL  EXPLANATION  OF  SQUARE  ROOT. 

Find  the  square  root  of  2025. 


402 


OPERATION. 

2025|40  +  5 
1600 


2x40=80 
_5 
85 


425 
425 


Explanation, — Suppose  you  wish  to 
lay  a  square  floor  containing  2025  sq. 
ft.  You  want  to  know  its  dimensions. 
Cut  a  piece  of  paper  3  inches  square. 
As  near  as  we  can  determine  by  in- 
spection 1600  (40-)  sq.  ft.  is  tlie  largest 
floor.      Let  your  paper  represent  this 

square  floor  and  label  as  shown  in  the  accompanying  figure.     There 

are  still  425  sq.  ft.  to  be  built 

on.     By  adding  strips  of  the 

same  width  to  either  2  or  4 

sides   of  a  square,    we   shall 

preserve  the  square  form.     It 

is  easier  to  add  to  2  sides.    The 

strips  put  on  will  be   of  the 

same    length    as    the  square 

already  made,  or  40  ft.;  mak- 
ing the   2  strips  80  ft.  long. 

Dividing  the  area  425  sq.  ft., 

which  is  to  be  put  into  these 

additions,    by    their    length, 


40  ft. 

5  ft. 

•20U  sq.  ft. 

25 
.<q.  ft. 

1600  sq.  ft. 

00 

O 

o 

40  ft. 


5  ft. 


240  CALIFORNIA   SERIES. 

gives  their  width  5  ft.  Cut  two  pieces  of  paper  each  4  in.  by  )4  in- 
and  lay  by  tlie  square  as  shown  above,  labeUng  each  properly. 
But  a  square  corner  5  ft.  each  way  must  be  put  on  to  complete  the 
square.  How  wide  must  you  cut  the  paper  for  this  square ?  The 
whole  length  of  the  3  additions  is  85  ft.;  width,  5  ft.;  area,  425  sq.  ft. 

APPLICATION  OF  SQUARE  ROOT. 

EXERCISE   253. 

To  illustrate  the  following  examples,  draw  figures  and 
label  them. 

1.  Find  the  side  of  a  square  field  whose  area  is  1024  sq. 
rd. 

2.  An  orange  orchard,  containing  3364  trees,  has  the 
same  number  of  rows  that  there  are  trees  in  a  row.  How 
many  rows  has  it? 

3.  A  farmer's  ranch,  containing  640  acres,  is  in  square 
form;  how  many  rods  around  it? 

4.  I  have  a  garden  66  ft.  X  148^  ft.  What  is  the  side  of 
a  square  field  equal  in  area? 

";^  Find  the  dimensions  of  a  rectangvdar  field  containing 
3200  sq.  rd.,  and  twice  as  long  as  broad. 

J&7  What  is  the  side  of  a  square  field  equal  in  area  to  a 
triangular  field  containing  4096  sq.  rd.  ? 
^1     (^  Mr.  A  has  a  field  12  rods  square,  and  IMr.  B  a  square 
field  containing  12  sq.  rd.     What  is  the  difterence  in  their 
area? 

^  How  many  rods  of  fence  will  inclose  a  square  field  of 
4  acres  ? 

8  Metric.  Find  the  length  of  a  fence  which  will  inclose  a 
square  farm  of  23  hektares. 

^^  A  has  a  square  field  of  10  acres;  B  a  rectangular  field 
of  10  acres,  4  times  longer  than  broad.  Which  field  will  be 
the  cheaper  to  fence  at  $2.25  a  rod? 

9  Metric.  I  have  a  field  387.5  meters  long  and  174.8 
meters  wide.     My  neighbor  has  a  square  field  of  the  same 


ARITHMETIC. 


241 


area.  How  much  more  will  it  cost  to  inclose  my  field  than 
my  neighbor's  at  1.25  francs  per  meter  of  fence? 
^.  If  it  costs  $425  to  fence  a  field  72  rd.  X  98  rd.,  what 
will  it  cost  to  fence  a  square  field  of  the  same  area? 
*^.  What  are  the  dimensions  of  the  largest  possible 
square  table  that  can  be  made  from  a  rectangular  board 
128  in.  long  and  32  in.  wide? 


CUBE  ROOT. 


What  is  the  contents  of  a  cube  whose  edge  is  4  inches  ? 
The  product  of  a  number  used  three  times  as  a  factor 
is  called  the  cube  of  the  number. 

The  number,  or  factor,  is  the  cube  root  of  the  product. 

Thus,  64  is  the  cube  of  4 ,'  4  is  the  cube  root  of  64- 

The  cube  is  indicated  thus:  4^,  <^^  ^^ 

The  cube  root,  thus:  f  64,^^7,  f/l£o  ;  or  64^,27^,125^- 


EXERCISE  254. 

1^                 ^27 

m 

10^ 

^  .3^ 

1^                ^64 

{W 

(iV)^ 

A' 

2^                  h' 

{iY 

.r 

0.27'^' 

8^                 125^ 

(t¥5)^ 

1000^ 

.008^ 

3^             {^y 

{irY          . 

.001« 

.064^ 

4«                  {lY 

(H)^ 

.2^ 

(  1   "1^ 

\12o/ 

Find  the  cube  of 

25. 

r         2on      I 

25^= )  2X20X5    ,X^ 


J 


r  20^=8000=tens.' 

r20=-|  2  X  20' X  5^4000=2  X  tens' Xu. 
'  [       20X5'=  500=t'nsX  units.' 

r       20'  X  5=2000=t'ns'  X  units. 
2  X  20  X  5'=1000=2  X  tens  X  u.' 
5^=  125=units.' 


L  5= 


16— A 


15625 


242 


CALIFORNIA    SERIES. 


Hence  to  cube  a  number  of  two  figures,  add  tens^,  3  X 
tens'  X  units,  3  X  tens  X  nnits^  and  units^ 


To  extract  the  cube  root  of  a  number. 

Find  the  cube  root  of  15625. 


20'= 
3  X  20'=  1200 


FULL    OPERATION. 

15625.(20  +  5 
8000 
762l 
6000 


3X20X5' 


1625 
1500 


Explanation. — 15625   = 

tens^  +  3  X  tens'-^  X  units  + 
3  X  tens  x  units^  +  units^. 
The  largest  tens^  in  15625  is 
8000  =  20^   The  remainder, 

3  X  tens2       ) 


CONTRACTED    OPERATION. 

15625.(25 

8 


3X20'=1200     7  625 
3X20X5=  300  ' 


f;2_ 


25 


1525 


7625 


7625  =  -(  3  X  tens  X  units  >  X  units. 
(  nuits2  )      . 

3  X  tens2  (20^)  =  1200.  This 
being  the  largest  part  found 
in  7625,  dividing  7625  by 
1200  gives  units  5  (more 
nearly  6,  but  allowance 
inust  be  made  for  the  other 
parts  in  7625)  and  1625 
over.  1625  contains  3  x 
tens  X  units'-^,  or  3  x  20  x  5"-^ 
=  1500,  and  units^,  or  5^,  = 
125. 


The  O's  may  be  omitted,  as  shown  in  the  contracted 
work;  and,  instead  of  multiplying  1200  (3  X  tens'),  300 
(3 X tens X units),  and  25  (units')  by  5  separately,  we  mul- 
tiply their  sum  by  5. , 

In  cubing  a  number,  as  56.8, 
.8"'=  .512         We  see  that  the  cube  of  each  figure 

6^    ^=       216.  occupies  3  places.     Hence  point 

5'    .    =125       .  off  the  number  whose  cube  root 

is  to  be  taken, into  groups  of  3  figures  each,  commencing  at 
the  decimal  point.  Each  group  will  be  used  in  finding  the 
figure  whose  cu))e  is  in  it.  Make  full  periods  at  the  right 
of  the  decimal  by  annexing  1  or  2  O's. 


ARITHMETIC. 


84027.672^=? 


3x40-=4800 

3X40X     3=   360 

3'=        9 


5169 

3X430^=554700 

3X4*30X8=    10320 

8^=  64 


565084 


84^0  2  7/6  72^(43.8 
64 


2002 


15507 


4520672 


4520672 


EXERCISE   255.    (Written.) 


Find  the  cube  root  of: 


1. 

195112. 

aa. 

46656. 

23. 

.004096 

2. 

262.144 

13. 

7_2iL 
4096 

24. 

13.824 

3. 

.830584 

14. 

343 
5  12 

25. 

970299. 

4. 

512 
7  2  9 

15. 

279726.264 

26. 

3| 

5. 

17576. 

16. 

54872. 

27. 

91  0 

2  7 

6. 

175.76 

17. 

12.167 

28. 

15| 

7. 

81fV 

18. 

1.2167 

29. 

10  00 
1331 

8. 

166.375 

19. 

91125. 

30. 

39.304 

9. 

74.088 

20. 

1.728 

31. 

1577635. 

10. 

.117649 

21. 

2197. 

32. 

2  to  2  decimal  places 

11. 

531442. 

22. 

.005832 

33. 

7  to  2  decimal  places 

PRACTICAL  EXPLANATION  OF  CUBE  ROOT. 
Find  the  cube  root  of  10648. 


OPERATION. 


20^  = 
3x  20"'  =  1200 
3X20X2  =    120 

2^= 4^ 

1324 


10.6481  20+2 
8000 


2648 


2648 


244 


CALIFORNIA   SERIES. 


Explanation, — Suppose  we  have 
to  cut  a  cubical  block  of  stone  to 
contain  10648  cu.  in.  AVe  wish  its 
dimensions. 

The  largest  cube  that  can  be  deter- 
mined by  inspection  contains  8000 
cu.  in.,  or  20^  Its  edge  will  be  20 
in.  2648  cu.  in.  remain  to  add  to 
the  block.  Draw  on  the  board  a 
cube  similar  to  the  figure  here  and 
label  it  the  same. 


8000  cu.  in. 


Since  a  cube  has  six  equal  faces, 
we  may  cover  them  all  with  blocks 
of  equal  width  and  preserve  the 
cubical  form ;  or  better,  three  adja- 
cent faces.  These  3  additions  will 
be  20  in.  by  20  in.  or  each  have  400 
sq.  in.  in  their  face,  making  1200 
sq.  in.  for  the  surface  of  the  three. 
They  can  contain  2648  cu.  in. 
Therefore,  their  thickness  will  be 
2648^1200  =  2  in.  with  248  cu.  in. 
over.  Draw  these  additions  on  the 
board  and  label. 


2400  cu.  iu. 


Three  oblong  pieces  20  in.  long, 
2  in.  wide,  and  2  in.  thick,  con- 
taining in  all  240  cu.  in.,  must  be 
added.  Draw  and  label.  Lastly, 
a  small  cube,  whose  edge  is  2  in., 
contents  8  cu.  in.,  must  be  add- 
ed. Draw  and  label.  The  cube 
is  now  complete.  The  addi- 
tions contain  2648  cu.  in.,  using 
all  the  material. 

Draw  the  completed  cube  rep- 
resenting the  additions  as  shown 
in  the  figure  on  the  next  page. 


248  cu.  ill. 


ARITHMETIC. 


245 


The  teacher  should  illustrate 
each  step  by  the  blocks;  and 
extend  the  work  to  a  second  set 
of  additions,  making  three  fig- 
ures in  the  root. 


10648  cu.  in. 


PRACTICAL  APPLICATION  OF  CUBE  ROOT. 

EXERCISE  256. 

1.  Find  the  dimensions  of  a  cubical  box  which  contains 
9261  cu.  in. 

2.  Find  the  dimensions  of  a  cubical  tank  which  holds 
1000  gallons. 

'"8.  The  area  of  one  of  the  faces  of  a  cubical  box  is  576  sq. 
in.     How  much  will  it,  hold  ? 

4.  How  many  gallons  will  a  tank  hold,  of  cubical  form, 
the  area  of  whose  faces  is  3750  sq.  in.? 

5.  What  is  the  surface  of  a  cube  containing  2744  cu.  in.? 

6.  What  are  the  dimensions  of  a  cubical  box  containing 
I  as  much  as  one  whose  edge  is  4  feet  ? 

7.  A  certain  cubical  tank  contains  1728  cu.  in.  What 
will  a  tank  whose  edge  is  twice  this  contain? 

8.  A  cubical  cistern  holds,  when  full,  4238  kilograms  of 
water.     What  are  its  dimensions? 

9.  The  roof  of  a  certain  building  is  225  meters  by  14.2 
meters,  horizontal  dimensions:  2.^  centimeters  of  rain  just 
fill  a  cubical  cistern  into  which  the  roof  drains.  Find  the 
dimensions  of  the  cistern. 


246 


CALIFORNIA   SERIES. 


MENSURATION. 

LINES,  ANGLES,  AND   SUEFACES. 


'5}  To 


a 
< 


f=.2 


•ri 

^;r^ 

O 
S! 

■^i>^ 

ir 

b 

Hl| 

H 

^-^^liii'puv 

^'^ 


o  2 


.::      a  P< 


or  Width, 


ARITHMETIC. 


24ri 


Regular  Polygon. 
r  i  m  e  t  e 


Circle. 


In  right-angled  figures  the  ividth  and  length  are 
Observe. -<!      sides  of  the  figure. 

In  slanting-line  figures  the  ividth  is  not  a  side. 

Draw  these  figures  on  your  slate  until  they  are  familiar. 

Write  definitions  of  each  term  given  above,  from  the 
appearance  of  the  figure  and  the  directions.  Write  exam- 
ples of  lines,  angles,  and  surfaces  similar  to  the  above,  that 
you  see  in  the  room  or  remember.  Into  what  kind  of  fig- 
ures does  a  diagonal  divide  a  rectangle?  How  does  the 
radius  of  a  circle  compare  in  length  with  the  diameter? 


Draw  on  your 
slate  two  lines 
meeting  so  as  to 
make  a  square 
corner.  Have  one 
line  4  in.  long; 
the    other,    3    in. 

Now  draw  a 
line  between  their 
free  ends  and   measure   it' 

If  your  drawing  has  been 
exact  the  third  line  will  be 
just  5  in.  long.  Make  these 
lines  very  heavy  and  build 
squares  upon  them  as  shown 
in  this  figure.  Divide  each 
line  into  inch  parts  by  dots, 


Relations  of 
the  sides  of  a 
right  -  angled 
triangle. 


yr             Base.                ~ 

i i           i 

248  CALIFORNIA   SERIES. 

make  inch  squares  by  cross  lines,  and  count  the  number 
of  inch  squares  in  each  hirge  square. 

Notice  how  the  number  in  the  hypotenuse  square  cor- 
responds with  that  in  both  the  other  squares  put  together. 

The  correspondence  found  here  is  always  true  of  right- 
angled  triangles;  namely: 

/     The  square  of  the  hypotenuse  is  the  sum  of  the  squares  of 
the  other  sides. 

The  difference  of  the  squares  of  the  hypotenuse  and  either 
side  is  the  square  of  the  other  side. 

Find,  by  these  laws,  what  should  be  the  hypotenuse  if 
the  base  is  8  in.  and  the  perpendicular  6  in.  Draw  on  the 
board  and  see  if  it  is  true.  Try  the  same  with  12  and  9 
in.;  with  16  and  12  in. 

EXERCISE  257.    (Written.) 

To  illustrate  the  following  examples,  draw  figures  and 
label  them. 

1.  Perpendicular,  10  ft.     Base,  10  ft.     Hypotenuse? 

2.  Base,  15  ft.     Hypotenuse,  20  ft.     Perpendicular? 

3.  Perpendicular,  18  ft.     Hypotenuse,  25  ft.     Base? 

4.  Distance  diagonally  across  a  floor  30x40  ft.? 

5.  Distance  diagonally  across  a  blackboard  8X3  ft.? 

6.  Length  of  a  ladder  to  reach  the  eaves  of  a  building  22 
ft.  high,  the  base  of  the  ladder  being  placed  6  ft.  from  the 
building  ? 

7.  If  you  draw  the  preceding  ladder  out  3  ft.  at  the  bot- 
tom, how  high  will  it  reach? 

8.  What  length  of  rope  will  reach  from  the  top  of  a  24- 
foot  pole  to  the  ground  on  the  opposite  side  of  a  street  60 
feet  wide? 

9.  A  rope  250  feet  long  was  stretched  from  one  bank  of 
a  river  to  the  top  of  a  pole  65  feet  high  on  the  opposite 
bank;  how  wide  was  the  river? 


I 


I 


ARITHMETIC.  249 

10.  A  tree  broken  off  14  feet  above  ground  rested  on  the 
ground  14  feet  from  tlie  stump.     How  tall  was  the  tree? 

11.  Find  the  distance  from  the  upper  corner  to  the  oppo- 
site lower  corner  of  a  room  40x30x12. 

12.  A  pole  is  held  vertical  by  wires,  one  of  w^hich  is  82 
feet  long,  stretched  from  the  top  of  the  pole  to  the  top  of  a 
stake  10  feet  high  and  36  feet  from  the  pole.  How  high  is 
the  pole? 

13.  What  is  the  length  of  a  path  diagonally  across  a  10- 
acre  square  field  ? 

14.  Distance  from  the  center  of  the  a])Ove  field  to  the 
center  of  a  side  ? 

15.  Diagonal  of  a  cube  containing  729  cu.  in.? 

16.  What  is  the  side  of  a  square  field  whose  diagonal  is 
15  rods?     Its  area? 

17.  A  ladder  28  feet  long  placed  in  a  street  reaches  the 
top  of  a  building  18  feet  high  on  one  side  and  one  15  feet 
high  on  the  other.     How  wide  is  the  street? 

18.  Two  vessels  sail  from  the  same  point,  one  north  58 
miles,  and  the  other  west  72  miles.  How  far  apart  are 
they? 

19.  Find  the  longest  straight  stick  you  can  put  ii\to  a  box 
2^  ft.  long,  1^  ft.  wdde,  and  12  in.  deep. 

20.  What  is  the  length  of  the  rafters  of  a  building  hav- 
ing a  gable  roof,  the  building  being  36  ft.  wide,  the  eaves 
20  ft.,  and  the  ridge-pole  30  ft.  from  the  ground? 


SUEFACE  AEEAS. 

The  parallelogram  is  the  basis  for  computing  areas,  its 
area  being  the  length  times  the  width. 

All  triangles  are  \  the  size  of  a  paral-  XT""^^---.^ 
lelogram  of  equal  length  and  width.  Thus,      \       ^'""•"-^^^^ 

Figures  of  equal  or  parallel  sides  having  more  than  4 


250 


CALIFORNIA   SERIES. 


sides,  and  irregular  figures,  are  divided  into  triangles  to 
find  their  areas.     As  fig.  6,  p.  246,  and  fig.  7,  p.  247. 

A  circle  may  be  regarded  as  made 
up  of  a  very  great  number  of  equal 
triangles  having  their  vertices  at  the 
center  and  their  bases  forming  the  cir- 
cumference. The  radius  of  the  circle, 
therefore,  is  the  uniform  width,  and  the 
circumference  the  continuous  bases  of 
the  triangles. 

When  the  sides  of  a  regular  polygon  (see  p.  247)  are  very 
great  in  number  (infinite)  the  perimeter  becomes  a  circum- 
ference and  the  apothem  a  radius. 


Area  of 


Parallelograms  =  length  x  width. 

(  base ) 

1  Triangles  =  )2  length-    perim...  .    - 
V  (  circuni. . .    ) 

Circumference  =3. 1416  (3t)  x  diameter. 


X  width 


.  .  .width, 
.apothem. 
.  .  .radius. 


EXERCISE  258.    (Written.') 

To  illustrate  the  following  examples,  draw  figures  and 
label  them. 

1.  A  triangular  field  is  20  rods  long  and  18  rods  wide; 
what  is  its  area  ? 

2.  A  field  has  two  parallel  sides  25  and  o5  rods  long, 
respectively,  the  distance  between  them  being  13  rd.  What 
is  the  area  of  the  field  ? 

3.  What  is  the  circumference  of  a  circular  pond  whose 
radius  is  11  rods?     Its  area? 

4.  What  is  the  radius  of  a  circle  equal  in  area  to  a  tri- 
angle 1,3X10  ft.? 

5.  A  horse  is  tied  to  a  stake  by  a  40-foot  rope.  What 
area  of  ground  can  lie  graze  over? 

6.  A  circular  map  of  the  Eastern  Hemisphere  is  to  be  3 
feet  in  diameter.     What  surface  will  it  cover? 


ARITHMETIC.  251 

7.  A  regular  6-sided  room  has  its  side  6  feet  long  and 
the  distance  from  the  center  of  the  room  to  a  side  is  5.196 
feet.     How  many  sq.  yd.  of  carpet  will  cover  it? 

8.  The  area  of  a  triangular  field  is  135  sq.  rd.,  and  its 
length  18  rd.;  find  its  width. 

9.  Find  the  area  of  a  right-triangle,  two  of  whose  sides 
are  equal,  and  the  third  is  72  feet. 

10.  I  have  256  sq.  ft.  of  boards.  If  laid  in  a  floor  of  tri- 
angular form  12  ft.  wide,  how  long  will  it  be? 

11.  What  is  the  area  of  a  board  18  ft.  long,  whose  ends 
are  respectively  12  in.  and  6  in.? 

12.  AVhat  is  the  distance  through  a  tree  that  girts  12  ft. 
6  in.? 

13.  The  radius  of  a  circle  is  10  feet;  find  the  radius  of  a 
circle  containing  9  times  its  area;  4  times. 

14.  A  cow  is  one  day  tied  to  the  top  of  a  stake  5  ft.  high 
by  a  rope  20  ft.  long;  on  the  next  day  she  is  tied  to  the 
bottom  of  the  same  stake  by  the  same  rope.  Find  the  dif- 
ference in  the  areas  over  wdiich  she  can  graze. 

15.  What  will  it  cost  at  $2  a  rod  to  fence  a  circular  plot 
of  land  containing  1  acre  ? 

16.  Find  the  cost  of  a  triangular  field  72X54  rods^at  $125 
an  acre. 

17.  At  $85  an  acre,  and  $1.75  a  rod  for  fencing,  what  are 
my  expenses  in  the  purchase  and  fencing  of  a  field  having 
two  parallel  sides  108  and  144  rods  long,  respectively,  the 
distance  between  them  being  96  rods,  and  the  other  two 
sides  being  97.67  rods  each? 

18.  A  gravel  walk  around  a  rectangular  grass  plot  12  ft. 
X8  ft.  is  2  ft.  wide;  what  is  its  area? 

19.  How  many  times  will  a  carriage  wheel  4  ft.  in  diam- 
eter turn  around  in  going  1  mile  ? 

20.  A  square  field  contains  31.5  acres;  what  is  the  length 
of  its  diagonal?  What  is  the  circumference  of  a  circular 
field  of  the  same  area  ? 


252 


CALIFORNIA   SERIES. 


SOLIDS— SURFACES  AND  CONTENTS. 


Prisms. 


Pyramid. 


Cone. 


/ 

Upper  Base.   / 

1. 

/l 

ower  Base. 

/ 

Cylinder. 


Frustum 

of 
Pyramid. 


i 


\ 


Up. 
Base. 


L. 

Base. 


V 


Sphere. 


Frustum 
of  Cone. 


The  distance  from  the  top  to 

the  bottom,  measured  by  a  plumb 

line,  or  actual  height  (altitude) 

of  jigs.  4,  5,  7,  a]id  8  is  shorter 

^,  .   tha7i    the    distance    measured 

Observe.  <    ,         ,,       . , 
down  the  side. 

The  distance  down  the  side 

perpendicularly  is    called   the 

slant  height,  and  is  used  in 

^finding  the  areas. 


Draw  these  figures  on  your  slate  until  they  are  familiar. 
Name  any  surfaces  on  these  that  are  like  the  surfaces  on 
p.  246. 


ARTTinrETrc. 


253 


Write  examples  of  these  solids  that  you  see  in  the  room 
or  remember.     Turn  fig.  7  upside  down  for  examples. 

The  surfaces  of  all  the  preceding  solids  are  like  the  plane 
surfaces  on  p.  246.     Thus, 


The  side  surface  of  a  prism  or  cylinder  is  seen  to  be  a 
parallelogram  whose  length  is  the  distance  around,  and 
whose  width  is  the  height  of  the  prism  or  cylinder. 

Prove  this  by  cutting  a  paper  rectangle  and  folding  to 


make  the  above  figures. 

c 


The  side  surface  of  a  regular  pyramid  or  cone  is  seen  to 
be  made  up  of  triangles  whose  length  is  the  distance  around 
the  base  and  whose  height  is  the  slant  height  of  the  pyra- 
mid or  cone.  d 

D 


254  CALIFORNIA   SERIES. 

The  side  surface  of  any  regular  frustum  is  made  up  of 
trapezoids. 

Upper  and  lower  bases  are  circles  or  polygons. 

The  surface  of  a  sphere  is  4  times  that  of  a  circle  having 
an  equal  diameter. 

A  rectangular  prism  (e.  g.  a  room)  is  the  basis  for  com- 
puting the  contents  of  solids.  Its  contents  is  the  product 
of  three  dimensions;  length,  width,  and  height;  or  the  area 
of  the  base  (length  X  width)  multiplied  by  the  height. 

This  gives  the  law  for  prisms  and  cylinders. 

A  pyramid  or  cone  is  found  to  contain  -J  as  much  as  a 
prism  or  cylinder  of  equal  base  and  height;  a  sphere  |  as 
much  as  a  cylinder  of  equal  circumference  and  a  height 
equal  to  the  diameter  of  the  sphere.     Hence, 

^  - .    ^      =  Area  of  base  X  height. 
Cylmder  ,) 

:     y^^^^^     .  ^  Area  of  base  X  i  height. 
Contents  oii  Cone         ) 

Sphere  =  Great  circle  X  |  diameter. 

Frustum  =  (Sum   of  areas   of   bases  -f-  the 

square  root  of  their  product)  X  i  height. 

EXERCISE  259.    (Written.) 

Draw  figures  and  label  them. 

1.  Find  the  contents  of  a  box  whose  length,  width,  and 
depth  are,  respectively,  4  ft.,  3  ft.,  and  2  ft. 

2.  Find  its  surface. 

3.  Find  the  number  of  square  feet  necessary  to  make  a 
piece  of  stovepipe  2-|  feet  long  and  5  in.  in  diameter. 

4.  Find  the  amount  of  tin  necessary  to  make  a  tin-pail 
cylindrical  in  form,  6  in.  in  diameter  and  8  in.  deep,  with- 
out a  cover. 

5.  How  many  quarts  will  the  pail  hold? 

6.  Find  how  much  water  can  be  put  into  a  tin-pail,  10  in. 


/^ 


ARITHMETIC.  255 

deep,  like  a  frustum  of  a  cone  in  form,  whose  bottom  is  8 
in.  across,  and  top  12  in.  across. 

7.  How  many  sq.  ft.  of  tin  in  the  pail  described  in  the 
last  example,  without  cover? 

8.  A  cylindrical  bottle,  containing  1  quart  of  ink,  is  3  in. 
in  diameter;  how  deep  is  it? 

9.  A  draughtsman  puts  a  map  of  the  world  on  a  globe  12 
in.  in  diameter;  what  area  does  it  cover? 

10.  If  your  ink  well  is  1  in.  across  and  1  in.  deep  how 
many  times  can  it  be  filled  from  a  quart  bottle  ? 

11.  A  conical  wood  pile  is  6  ft.  high  and  12  ft.  in  diam- 
ete^at  the  base;  how  many  cords? 

12.  How  many  bushels  of  oats  in  a  conical  pile  2  ft.  high 
and  12  ft.  around  it  at  the  base? 

13.  Anticipating  rain,  the  above  pile  is  covered  with  tent- 
cloth  ;  how  many  square  yards  ? 

^Hh  Tind  the  depth  of  a  cylindrical  tank  that  holds  20 
gallons,  and  is  18  in.  in  diameter. 

ISy  If  the  above  tank  has  a  conical  top  4  in.  high,  how 
many  more  gallons  can  be  put  in  ? 

>•    16.  The  glass  tank  of  a  lamp  is  spherical  in  shape  and 
^n.  in  diameter  on  the  inside;  how  much  oil  will  it  hold? 

^L  If  a  5-gallon  oil  can  is  10  in.  square  on  the  bottom 
how  deep  is  it? 

18.  What  is  the  difference  in  the  number  of  square  feet 
of  lumber  necessary  to  make  the  sides  of  a  room  16  ft.  long, 
12  ft.  wide,  and  10  ft.  high,  and  one  of  circular  floor  con- 
taining the  same  area  and  of  the  same  height? 

iV  Find  the  number  of  cu.  ft.  inclosed  by  a  barn  60  ft. 
long,  40  ft.  wide,  and  20  ft.  high,  with  a  pyramidal  roof  8 
ft.  high,  all  inside  measurements. 

20.  How  many  cu.  ft.  of  wood  are  in  a  log  20  feet  long 
and  14  in.  in  diameter. 

21.  The  earth's  diameter  is  about  8000  miles;  what  is 
its  area  ?     Its  volume  or  bulk  ? 


256  CALIFORNIA   SERIES. 

22.  At  28  ct.  per  cu.  ft.  what  is  the  cost  of  a  stone  wall  28 
in.  thick  at  the  base  and  18  in.  at  the  top,  4  ft.  high  and  3G 
rd.  long  ? 

23.  The  above  wall  is  laid  on  a  foundation  of  Portland 
cement  4  inches  wider  than  the  base  of  the  wall,  and  8  in. 
deep.     What  is  the  cost  of  the  foundation  at  32  ct.  a  cu.  ft.  ? 

24.  How  many  cu.  ft.  in  a  regular  8-sided  post  10  feet 
high,  the  length  of  one  side  being  3  in.,  and  the  distance 
through  it  7.24  in.  ? 

EXERCISE  260.    (Oral.) 

1.  Area  of  a  triangular  field  10  rd.  long  and  8  rd.  wide? 

2.  AVidth  of  a  triangular  field  containing  1  A.,  length  20 
rd.? 

3.  Width  of  a  rectangular  field,  with  dimensions  as  in 
Example  2? 

4.  Number  of  cu.  ft.  in  a  conical  pile  G  ft.  high  and  7  ft. 
across  at  the  base? 

5.  Area  of  a  square  field  whose  diagonal  is  20  rd.? 

6.  Radius  of  a  circle  whose  circumference  is  6-f-  ft.? 

7.  Area  of  a  field  having  two  parallel  sides  40  and  30 
rd.  respectively^  and  width  10  rd.? 

8.  How  many  times  will  a  wheel  of  3^  ft.  radius  turn 
around  in  going  4  rods  ? 

9.  Length  of  rafters  on  a  barn  whose  gable  end  is  32  ft. 
wide  and  the  roof  12  ft.  high? 

10.  Number  of  sq.  in.  of  material  to  make  a  2-ft.  length 
of  7  in.  stove  pipe,  allowing  1  in.  for  lapping? 

11.  Cost  of  fencing  the  field  in  Example  3  at  $2  a  rod? 

12.  Height  of  a  pyramid  containing  144  cu.  in.,  the  area 
of  whose  base  is  36  sq.  in.? 

13.  Height  of  a  cone  of  the  same  measurement  as  the 
preceding  pyramid  ? 

14.  Number  of  cu.  yd.  of  gravel  necessary  to  cover  a  walk 
3  ft.  wide,  54  ft.  long,  and  3 in.  deep? 


ARITHMETIC.  257 


MISCELLANEOUS  PROBLEMS. 

EXERCISE   261. 

1.  Spent  \  of  my  money  for  a  watch,  g\  of  the  remainder 
for  a  chain,  \  of  what  then  remained  for  a  suit  of  clothes, 
and  "3^  of  the  rest  for  a  pair  of  shoes,  when  I  had  $150  left; 
what  had  I  at  first? 

2.  Imported  7  casks  of  brandy,  30  gal.  each,  duty  $2  per 
gal.,  charges  $27;  sold  the  whole  for  $1714.27^,  gaining 
A2\%  on  the  whole  cost;  what  was  the  cost  of  the  liquor 
per  gal.  at  the  foreign  port? 

3.  A  man  dying  leaves  in  the  savings  bank  for  his  16- 
year  old  son  such  a  sum  of  money  as  shall  amount  to  $5000 
when  the  son  is  21.  If  the  bank  adds  the  interest  to  the 
principal  every  half  year,  how  much  money  must  be  left  in 
the  bank?    Interest  at  6%. 

4.  My  agent  sells  for  me  800  bbl.  flour  at  $4.75,  commis- 
sion If  %,  and  buys  sugar  at  6^  cents  a  ife.,  commission  2%] 
what  is  the  whole  commission,  and  how  many  lb.  of  sugar 
do  I  receive? 

5.  A  rectangular  field  of  4^  A.,  whose  breadth  is  f  its 
length,  is  surrounded  by  a  close  board  fence  8^  ft.  high, 
with  8-foot  posts  5  in.  square  and  8  ft.  apart,  and  two  rows 
2-by-4  scantling  around  the  field.  If  the  lumber  cost 
$440.44,  what  was  the  price  per  M  ? 

6.  Find  the  dimensions  of  a  rectangular  field  whose 
length  is  3  times  its  width,  and  whose  area  is  327.46  ares. 

7.  ^;tZlx41A=.? 

8.  Find  the  cost  of  carpeting  a  room  17  ft.  by  13  ft.  2  in. 
with  carpet  1  meter  wide,  at  85  cents  a  meter,  laid  length- 
wise. 

9.  Add    5.13875    miles    and    25.312    rods,  take   away 

147.3125  yards,  and  give  the  result  in  feet. 
17— A 


258  CALIFORNIA   SERIES. 

10.  A  horse  is  tethered  by  a  rope  12.4  meters  long;  what 
area  can  he  feed  ? 

11.  A  land  company  buys  36  acres  of  land  whose  breadth 
is  y^Q-  its  length,  and  divides  it  into  city  lots.  3  streets  80 
feet  wide  run  lengthwise,  and  2  streets  60  ft.  wide,  crosswise. 
The  lots  are  50  by  118^  ft.,  and  are  sold  at  $220.  How 
much  is  realized  by  the  sale  ? 

12.  Suppose  the  company  to  have  paid  $500  an  acre  for 
the  above  land,  and  $800  in  paving,  grading,  etc.;  what  is 
its  per  cent  of  profit  in  the  venture  ? 

13.  I  hire  money  at  7%  to  purchase  one  of  these  lots,  and 
after  the  lapse  of  15  mo.  I  sell  for  $500;  find  my  profit  %. 

14.  What  is  the  diameter  of  a  circle  whose  area  is  278.54 
ares? 

15.  A  grain  dealer  buys  3000  centals  of  barley,  i  of  which 
he  sells  at  a  gain  of  S%,  i  at  a  gain  of  12%,  ^  at  a  gain  of 
16%,  and  the  remainder  at  a  gain  of  20%.  Had  he  sold 
the  whole  at  a  gain  of  15%,  he  would  have  received  $54 
more.     Find  the  cost  per  cental. 

16.  A  commission  merchant  sold  cotton  cloth  on  1|% 
commission,  and  invested  the  proceeds  in  cotton  on  2|% 
Com.  If  his  commissions  amounted  to  $241.40,  what  sum 
was  received  for  the  cloth?     Sum  given  for  the  cotton? 

17.  Bought  a  note  for  f  its  face,  on  which  a  collector  ob- 
tained 25%  more  than  I  paid  for  it  and  charged  me  5%  for 
collecting.  If  I  realized  $75  by  the  transaction,  what  was 
the  fa,ce  of  the  note  ? 

18.  If  4  men  working  10  hr.  per  day  do  a  piece  of  work 
in  60  days,  how  many  men  will  it  take  to  do  twice  the  work 
in  40  days,  working  8  hr.  per  day? 

«  ?M^The  largest  circular  path  that  could  be  made  in  a  cer- 
tain square  garden  was  5^  rods  in  diameter;  what  was  the 
area  of  the  garden? 

20.  What  is  the  base  of  a  triangle  whose  area  is  §^.28 
ares,  and  whose  altitude  is  39.4  meters? 


ARITHMETIC.  259 

21.  Find  the  area  iu  hektares  of  a  piece  of  ground  1  mile 
square. 

22.  A  merchant  bought  goods  for  '$3600,  marked  them  at 
30%  advance,  and  finally  sold  them  at  10,  and  5  off  from 
the  marked  price  for  cash.     Find  his  selling  price. 

23.  How  large  a  draft,  payable  60  days  after  sight,  can 
be  bought  for  $502.25,  exchange  being  1%  and  interest  6%  ? 

24.  Express  as  a  decimal  ) ? i  lI\  i  / q     i' V \  vy  - • 

25.  A  and  B  together  own  540  acres  of  land  and  agree  to 
share  it  in  the  proportion  of  7  to  11.  AMiat  number  of 
acres  does  each  receive  ? 

26.  Find  the  surface  and  solidity  of  a  sphere  whose  diam- 
eter is  3.64  meters. 

27.  A  mechanic  agreed  to  work  80  days  on  condition 
that  he  should  receive  $1.75  and  board  for  every  day  he 
worked,  and  pay  75  cents  a  day  for  board  when  idle.  His 
earnings  were  $80;  how  many  days  did  he  work? 

28.  Says  A  to  B,  f  of  my  age  equals  |  of  yours.  The  sum 
of  their  ages  was  136;  find  the  age  of  each. 

29.  A  cylindrical  tank  is  3.8  meters  high,  and  diameter 
of  base  2.8  meters,  both  inside  measurements.  How  much 
water  will  it  hold  and  what  is  its  weight  in  kilograms? 

30.  Divide  448  A.  144  sq.  rd.  of  land  among  A,  B,  C,  and 
D,  so  that  A  shall  have  I  of  the  whole  +  4  A.  126  sq.  rd.; 
B  4-  of  the  remainder;  C  ^  of  what  then  remains;  and  D 
the  rest. 

31.  How  deep  a  ditch  3  ft.  wide  must  be  dug  around  a 
field  5  rods  square  that  the  earth  removed  may  raise  the 
surface  of  the  field  6  in.? 

32.  My  garden  is  43.6  meters  long  and  27.9  meters  wide. 
My  rain  gauge  registered  16  centimeters  in  the  late  storm. 
How  many  kilograms  of  water  fell  on  my  garden? 

SSi,  How  many  hogsheads,  of  63  gallons  each,  will  a  cyl- 
indrical tank,  10  ft.  in  diameter  and  10  ft.  deep,  hold  ? 


260  CALIFORNIA   SERIES. 

34.  What  is  the  vahie  of  $1  in  shillings  and  pence?  In 
francs  ? 

35.  A  rectangular  tank  holds  58248  liters  of  water:  two 
of  its  inside  dimensions  are  3.7  meters  and  3.42  meters; 
what  is  the  third  dimension? 

36.  Find  the  whole  cost  of  550  yd.  Brussels  carpeting  at 
$1.80  a  yard,  commission  for  purchasing  being  2^%,  draft 
^%^  and  $17  freight  prepaid. 

37.  A  certain  principal  at  a  certain  rate  amounts  to  $750 
in  3  yr.,  and  the  interest  is  i  of  the  princij^al.  Find  the 
principal  and  the  rate. 

38.  How  much  wood  in  a  pile  32.5  meters  long,  3.2  me- 
ters wide,  and  1.8  meters  high? 

39.  Two  men  dig  a  ditch  for  $53;  one  man  worked  3^ 
days  and  dug  14^  rd.  a  day;  the  other  worked  as  many 
days  as  the  first  dug  rods  per  day.  What  did  each  receive 
if  they  shared  in  proportion  to  the  time  worked  ? 

40.  A  and  B  furnish  capital  to  engage  in  business  and  C 
does  the  work  for  -^  the  profit.  A  contributes  $8000  and  B 
$10000.     They  gain  $5400.     Find  the  share  of  each. 

41.  If  52  men  can  dig  a  trench  355  ft.  long,  60  ft.  wide, 
and  8  ft.  deep,  in  15  days,  what  is  the  length  of  a  trench  45 
ft.  wide  and  10  ft.  deep,  which  45  men  can  dig  in  25  days  ? 

42.  At  what  price  must  cloth  that  cost  $3.50  a  yard  be 
marked  that  may  fall  20  per  cent  and  still  gain  20  per  cent 
on  the  cost? 

43.  Bought  8  cd.  6f  cd.  ft.  of  wood  at  $7.20  a  cord  and 
paid  in  equal  weights  of  butter  and  cheese  at  20  cents  a  lb. 
for  butter  and  12  cents  a  lb.  for  cheese.  How  many  lb.  of 
each  were  required? 

44.  Find  the  surface  of  a  cone  Avhose  altitude  is  3.8  me- 
ters and  diameter  of  base  2.28  meters. 

45.  Find  the  prime  factors  of  729,  336,  and  1836. 

46.  Paid  $2225  for  180  sheep  and  sold  them  for  $2675; 
what  should  I  gain  on  1500  sheep  at  the  same  rate? 


ARITHMETIC.  261 

47.  Find  the  g.  c.  f.  of  84,  336,  420,  and  504. 

48.  Write  the  38th  example  with  the  same  values  in  our 
measures  and  work  it,  giving  the  result  in  cords  and  feet. 

49.  What  is  f  of  an  acre  of  land  worth,  if  f  of  an  acre  is 
worth  $60? 

50.  A  tank  will  hold  420  gallons  and  is  |  full;  what  part 
full  is  it  if  87-|  gallons  be  added? 

51.  Bought  24  T.  4  cwt.  1  qr.  18  ft),  of  EngUsh  iron  at  3 
pence  per  lb.,  long  ton  weight,  and  sold  the  same  at  $142 
per  short  ton.     What  did  I  gain? 

52.  When  rain  falls  3  centimeters  in  depth,  how  many 
kilograms  have  fallen  on  a  garden  73.3  meters  long  by  38.18 
meters  wide? 


53.  $714.50.  Los  Angeles,  Aug.  28,  1885. 

For  value  received  I  promise  to  pay  H.  Miner,  or  order, 
Seven  Hundred  Fourteen  and  -j^^  Dollars,  on  demand,  with 
interest  at  12%  per  annum.  James  Towle. 

Find  the  amount  Mar.  17,  1886. 


54.  $534.00.  Los  Angeles,  Jan.  4, 1886. 
Six  months  after  date,  I  promise  to  pay  B.  Caldwell,  or 

order.  Five  Hundred  Thirty-four  Dollars,  with  interest  at 
^%  per  annum,  value  received.  W.  P.  Johnson. 

Discounted  at  a  Los  Angeles  bank  Mar.  17, 1886,  at  10%. 
Find  the  proceeds. 

55.  When  are  the  hour  and  minute  hands  of  a  clock  to- 
gether next  after  12  o'clock? 

56.  What  is  the  time  between  12  and  1  o'clock  when  the 
hour  and  minute  hands  are  equidistant  from  12  on  oppo- 
site sides? 

57.  I  buy  a  farm  for  $5000,  to  be  paid  for  in  5  payments; 
interest  at  10%  payable  annually.  The  payments  to  be 
0,  1,  2,  3,  and  4  years  from  date  of  purchase.  It  is  so  ar- 
ranged that  I  pay  exactly  the  same  amount  of  money  at  each 
payment.     What  is  the  equal  payment? 


262 


CALIFORNIA   SERIES. 


ABBEETIATIOI^S. 


A. 

Acre,  or  acres. 

gro. 

Gross. 

Acc't. 

Am't. 

Anal. 

Ans. 

Apr. 

Aug. 

Bal. 

bbl, 

bu. 

Account. 

Amount. 

Analysis. 

Answer. 

April. 

August. 

Balance. 

Barrel,  or  barrels. 

Bushel,  or  bushels. 

hdkf. 

bM. 

hr. 

in. 

Jan. 

1. 

lb. 

1.  c.  m. 

Handkerchief,  or  hand- 
kerchiefs. 

Hogshead,  hogsheads. 

Hour,  or  hours. 

Inch,  or  inches. 

January. 

Link,  or  links. 

Pound,  or  pounds. 

Least  common    multi 
pie. 

Meter, meters,  one  thou 
sand. 

bun. 
C. 

Bundle,  or  bundles. 
Cost. 

M. 

cd. 

Cord,  or  cords. 

Mar. 

March. 

cd.  ft. 

Cord  foot. 

Mdse. 

Merchandise. 

eh. 

Chain,  or  chains. 

mi. 

Mile,  or  miles. 

Co. 

Company, 

min. 

Minute,  or  minutes. 

Com. 

Commission. 

mo. 

Month,  or  months. 

Cr. 

Credit,  or  creditor. 

Mo. 

Monthly. 

c,  ct. 

Cent,  or  cents. 

No. 

Number. 

cu.  ft. 

Cubic  foot,  or  feet. 

Nov. 

November, 

cu.  in. 

Cubic  inch,  or  inches. 

Oct. 

October. 

cu.  yd. 

Cubic  yard,  or  yards. 

oz. 

Ounce,  or  ounces. 

cwt. 

Hundredweight. 

p. 

Page,  or  pages. 

d. 

Penny,  or  pence. 

P.  and  L. 

Profit  and  loss. 

da. 

Day,  or  days. 

pk. 

Peck,  or  pecks. 

Dec. 

December. 

Pop. 

Popular. 

deg. 

Degree,  or  degrees. 

pr. 

Pair,  or  pairs. 

do.,  ditto 

.  The  same. 

pt. 

Pint,  or  pints. 

doz. 
Dr. 

Dozen. 
Debtor. 

pwt. 

Pennyweight,    or    pen 
nyweights. 

far. 
Feb. 
ft. 
G. 

Farthing,  or  farthings. 

February. 

Foot,  or  feet. 

Gain. 

qr. 
qt. 
rd. 
Reed. 

Quire,  or  quires. 
Quart,  or  quarts. 
Rod,  or  rods. 
Received. 

gal. 

(Jallon,  or  gallons. 

rm. 

Ream,  or  reams. 

g.  c.  f. 

Cireatest  common   fac- 
tor. 

Sci. 
s. 

Science, 

Shilling,  or  shillings. 

gi. 

Gill,  or  gills. 

scr. 

Scruple,  or  scruples. 

gr. 

Grain,  or  grains. 

sec. 

Second,  or  seconds. 

gran. 

Granulated. 

Sept. 

September, 

ARITHMETIC. 


263 


S.  P»  Selling  price, 

sq.  eh.  S(juare  chain,  or  chains. 

sq.  ft.  Stjuare  foot,  or  feet. 

sq.in.  Square  inch,  or  inches, 

sq.  1.  Square  link,  or  links. 

sq.  mi.  Square  mile,  or  miles, 

sq.  rd.  Square  rod,  or  rods. 


sq.  yd. 

Square  yard,  or  yards 

T. 

Ton,  or  tons. 

vol. 

Volume,  or  volumes. 

wt. 

Weight. 

yd. 

Yard,  or  yards. 

yr. 

Year,  or  years. 

SIGISTS. 


+ 

Addition. 

Ratio,  or  Division. 

— . 

Subtraction. 

_ 

Equals. 

X 

Multiplication. 

% 

Dollars. 

^- 

Division. 

^ 

Cents. 

0 

Parenthesis. 

£ 

Pounds   (Eng.  money) 

% 

Per  cent. 

Equals  (used  in  propor 

. 

Decimal  Point. 

tion). 

<"c 

Account. 

@ 

At. 

o 

Degree. 

1 

Square  Root. 

1 

Minute  (circ.  measure). 

u 

The  same. 

II 

Second  (circ.  measure). 

Therefore. 

264 


CALIFORNIA   SERIES. 


GLOSSAEY. 


Some  of  the  terms  in  the  glossary  are  not  employed  in  the  body  of  the  book.  Such  as 
are  so  employed  are  indicated  by  the  figures  in  parentheses.  These  figures  refer  to  the 
page  on  which  the  subject  is  fii'st  noticed. 


Abstract  number,  (36),  a  number 
used  by  itself  without  reference 
to  any  particular  thing. 

Acceptance,  (229),  agreeing  to  the 
terms  of  a  draft  by  writing  one's 
name  across  the  face. 

Account,  (221),  a  record  of  debts 
and  credits. 

Accurate  interest,  (209),  interest 
for  days  computed  on  a  basis  of 
365  days  to  a  year. 

Addition,  (14),  the  process  of  put- 
ting two  or  more  numbers  to- 
gether into  one. 

Ad  valorem  duty,  (200),  a  tax  on 
the  value  of  imported  goods. 

Aliquot  part,  (115),  an  exact  divis- 
or, integral  or  fractional. 

Alloy,  (168),  a  mixture  of  two  or 
more  metals;  a  baser  metal 
which  is  mixed  with  a  finer,  as 
in  money. 

Altitude  or  actual  height,  (252), 
the  shortest  distance  from  the 
top  to  the  base  of  a  triangle, 
pyrandd,  cone,  or  frustum. 

Amount,  (14),  the  sum  of  two  or 
more  numbers;  (201),  in  Inter- 
est, the  sum  of  principal  and 
interest;  also,  a  sum  of  money. 

Analysis,  (172),  a  separation  into 
parts  for  the  special  treatment 
of  each. 

Angle,  (246),  the  difference  in  direc- 
tion of  two  lines  that  meet. 

Apothem,  (247),  the  distance  from 
the  center  of  a  regular  polygon 
to  the  central  point  of  any  side. 

Arc,  (146),  any  part  of  the  circum- 
ference of  a  circle. 

Are,  (131),  the  metric  unit  of  land 
measure. 

Area,  (129),  the  number  of  square 
units  in  a  surface. 


Arithmetic,  the  knowledge  of 
numbers  and  how  to  use  them. 

Assets,  (178),  the  actual  property 
of  a  person,  or  company. 

Average,  (235),  the  mean  of  two  or 
more  unequal  numbers. 

Avoirdupois,  (142),  the  weight  in 
common  use. 

Axis  of  the  earth,  (147),  a  straight 
line  joining  the  two  poles. 

Balance,  (221),  equality  of  weights 
or  numbers;  the  excess  of  one 
sum  of  money  over  another. 

Balance  sheet,  (226),  a  tabular 
statement  of  facts  showing  the 
condition  of  a  business. 

Bank,  (226),  an  establishment  for 
the  deposit,  exchange,  or  loan  of 
money. 

Bank  discount,  (218),  the  interest 
taken  by  a  bank  from  the  face 
or  amount  of  a  note,  for  paying 
it  before  it  is  due. 

Bankrupt,  one  declared  by  law  to 
be  unable  to  pay  his  debts. 

Base,  (246), in  geometrical  hgures, 
the  side  or  face  on  wliich  they 
stand;  (181),  in  Percentage, that 
number  of  which  another  is  a 
part  or  per  cent. 

Bill,  (119),  a  formal  statement  of 
goods  sold  or  services  rendered. 

Bill  of  exchange,  (229),  a  draft. 

Bond,  a  written  contract  for  the 
payment  of  a  sum  of  money  un- 
der given  conditions. 

Broker,  (190),  one  who  buys  and 

sells  for  another. 
Brokerage,  ( 190),  a  percentage  paid 

to  a  l)roker  for  doing  business. 

Cancellation,  (86),  the  division  of 
dividend  and  divisor  by  a  com- 
mon factor. 


ARITHMETIC. 


265 


Capital,  (178),  money  or  other 
property  by  means  of  which 
business  is  done. 

Carat,  (143),  a  24th  in  pure  gold  of 
the  entire  weight  of  a  mixture 
of  gold  and  baser  metals ;  thus, 
18  carats  tine  means  that  ||  of 
the  mixture  is  pure  gold. 

Cashier,  (228),  one  who  has  charge 
of  the  cash  and  cash  transac- 
tions of  a  banlc  or  company. 

Cental,  (142),  100  pounds. 

Centi-,  ( 127),  a  prefix  in  the  French 
metric  system  meaning  y^u  o^'. 

Certificate  of  deposit,  (227),  a  writ- 
ten statement  by  a  bank  that 
you  have  deposited  money  in  it. 

Chain,  (see  table,  p.  123). 

Check,  (227),  an  order  on  a  banlv 
for  money. 

Chord,  (14G),  a  straight  line  join- 
ing any  two  points  in  the  cir- 
cumference of  a  circle. 

Circle,  (seep.  146). 

Circulate,  (107),  the  repeating  fig- 
ures in  a  circulating  decimal. 

Circulating  decimal,  (107),  a  deci- 
mal fraction  in  which  the  same 
figure,  or  set  of  figures,  is  con- 
stantly repeated. 

Circumference,  (146),  the  bound- 
ing line  of  a  circle. 

Column,  (6),  a  vertical  line  of  num- 
bers. 

Commercial  discount,  (218,  219),  a 
deduction  on  the  face  of  a  bill, 
note,  or  other  writing  for  money. 

Commission,  (189),  a  percentage 
allowed  on  the  value  of  goods 
bought  or  sold,  money  collected, 
etc. 

Common  denominator,  (76),  one 
common  to  two  or  more  frac- 
tions. 

Common  factor,  (65),  a  whole  num- 
ber exactly  contained  in  each  of 
two  or  more  numbers. 

Common  multiple,  (68),  a  number 
which  exactly  contains  two  or 
more  whole  numbers. 

Company,  (194),  two  or  more  men 
uniting  in  some  business  or  en- 
terprise. 


Complex  fraction,  (91),  one  hav- 
ing a  fraction  or  mixed  number 
in  either  numerator  or  denomi- 
nator, or  both. 

Composite  number,  (63),  one  made 
up  of  factors. 

Compound  interest,  (217),  interest 
on  principal  and  unpaid  interest. 

Compound  number,  (122),  a  con- 
crete number  expressed  in  two 
or  ]nore  units. 

Compound  proportion,  (177),  an 
equality  between  a  simple  and  a 
compound  ratio. 

Compound  ratio,  (177),  the  indi- 
cated prodtict  of  two  or  more 
simple  ratios,  term  by  term. 

Concrete  number,  (36),  one  applied 
to  a  particular  object. 

Cone,  (252),  a  solid  whose  base  is  a 
circle  and  summit  a  point. 

Contents,  (129,  135),  the  number 
of  units  in  a  surface  or  solid. 

Corporation,  (201),  a  body  of  peo- 
ple authorized  by  law  to  do  busi- 
ness. 

Credit,  (223),  that  which  one  has 
paid. 

Creditor,  (223),  one  to  whom  a  debt 
is  owing. 

Cube,  (134),  a  solid  having  6  square 
faces;  (241),  the  product  of  a 
number  used  3  times  aS  a  factor. 

Cube  root,  (241),  one  of  the  3  equal 
factors  of  a  number. 

Customs,  (200),  taxes  on  imports  or 
exports. 

Cylinder,  (252),  a  straight  solid 
whose  bases  are  equal  and  par- 
allel circles. 

Days  of  grace,  (218),  3  days  that  a 
clebt  may  remain  unpaid  after 
the  time  due,  allowed  by  law  in 
many  states. 

Debt,  (223),  that  which  one  is  ow- 
ing. 

Debtor,  (223),  one  who  owes. 
Deci-,  (127),  a  prehx  in  the  French 
metric  system  meaning  ^-^  of. 

Decimal,  a  number  so  written 
that  each  character  is  tenfold 
greater  at  each  remove  from 
right  to  left. 


266 


CALIFORNIA   SERIES. 


Decimal  fraction,  (102),  that  part 

of  a  number  at  the  right  of  the 

decimal  point.    It  is  always  less 

than  a  unit. 
Decimal  notation,  (5),  the  art  of 

writing-  numbers  by  the  decimal 

scale,  or  scale  of  lO's. 
Decimal  point,  (5),  a  period  (.)  at 

the  right  of  units  in  a  decimal. 
Decimal  system,   the  method  of 

writing  numbers  in  decimals. 

Degree,  (146),  l-360th  of  a  circum- 
ference. 

Deka-,(127),  a  prefix  in  the  French 
metric  system  meaning  10. 

Denominator,  (72),  the  number  be- 
low the  line  in  a  fraction;  corre- 
sponds to  the  divisor  in  division. 

Diagonal,  (216),  a  line  joining  any 
two  corners  of  a  polygon  not 
lying  next  to  each  other. 

Diameter  of  a  circle  or  sphere, 
(U()),  a  straight  line  drawn 
through  the  center  and  termi- 
nating in  the  circumference. 

Difference,  (21),  the  result  obtain- 
ed by  taking  one  number  from 
another. 

Digits,  the  ten  symbols  of  the 
decimal  notation. 

Dimensions,  (254),  length  and 
breadth  of  a  surface,  or  length, 
breadth,  and  height  of  a  solid. 

Discount,  (219),  a  deduction  from 
the  face  of  a  debt;  (202),  in 
Stocks,  the  rate  the  market  value 
is  below  par. 

Dividend,  (43),  in  Division,  the 
number  to  be  divided;  (72),  in 
fractions,  the  numerator;  (202), 
in  business,  the  income  of  a 
stock  company. 

Division,  (43),  the  process  of  find- 
ing how  many  times  one  num- 
ber contains  another;  the  pro- 
cess of  separating  a  number  into 
ecpial  parts. 

Divisor,  (43),  the  number  by  which 
we  divide;  (72),  in  fractions,  the 
nominator. 

Draft,  (229),  a  written  order  l)y 
one  person  upon  another  to  pay 
money  to  a  third. 

Drawee  of  a  draft,  (see  p.  229). 


Drawer  of  a  draft,  (see  p.  229). 

Duty,  (199),  a  tax  on  imports  or  ex- 
ports. 

Equation,  a  statement  of  equality 
between  two  numbers  or  sets  of 
numbers. 

Equation  of  payments,  (233),  aver- 
age of  payments. 

Even  number,  one  having  2  for  a 
factor. 

Exact  divisor,  (63),  one  contained 
in  the  dividend  an  exact  num- 
ber of  times ;  may  be  integral  or 
fractional. 

Exact  interest,  (209),  interest  for 
days  computed  on  a  basis  of  365 
days  to  a  year. 

Exchange,  (228),  the  method  of 
making  payments  to  parties  at 
a  distance  by  drafts. 

Exponent,  (63),  a  figure  placed  to 
the  right  and  above  a  number, 
showing  how  many  times  the 
number  is  to  be  usecl  as  a  factor. 

Extremes,  (176),  the  first  and  last 
terms  of  a  proportion. 

Face,  (214),  the  sum  of  money 
mentioned  in  a  business  paper. 

Factor,    (35),    an    integral    exact 

divisor. 
Figures,  (5),  the  ten  symbols  of  the 

decimal  notation. 
Firm,  (178),  the  name  under  which 

a  company  does  business. 

Formula,  ( 17(')),  a  rule  expressed  by 
symbols  or  figures;  a  very  brief 
statement. 

Fraction,  (72),  an  indicated  divis- 
ion ;  one  or  more  equal  parts  of 
a  unit. 

Franc,  (157),  the  unit  of  French 
money. 

Frustum,  (252),  the  part  of  a  cone 
or  pyramid  left  after  cutting  otl" 
the  top  by  a  section  parallel  to 
the  base. 

Gain,  (185),  the  amount  by  which 
the  selling  i)rice  of  an  article 
exceeds  its  cost. 

Grace,  (218), an  allowance  of  3days 
made  by  some  states  for  the  pay- 
ment of  a  debt  after  the  set  time 
has  expired. 


ARITHMETIC. 


267 


Gram,  (145),  the  unit  of   metric 

weight    equal    to    15.43    grains 

Troy. 
Great  circle  of  a  sphere,  (254),  one 

that  cuts   the  sphere  into  two 

equal  parts. 
Greatest  common  factor,  (65),  the 

greatest  factor  common  to  two 

or  more  numbers. 
Greenback,  (221),  U.  S.  currency 

note,  now  payable  in  gold. 
Gross  -weight,  (200),  the  weight  of 

packed    goods,    including     the 

weight   of   the  boxes   or  other 

packing  material. 
Guarantee,  or  guaranty,  the  w^ar- 

ranting  by  another  of  the  pa}-- 

ment  of  a  debt. 

Hekto- ,( 127),  a  prefix  in  the  French 
metric  sj^stem  meaning  100. 

Horizontal,  (246),  parallel  to  the 

horizon. 
Hypotenuse,  (245),  the  longest  side 

of  a  right-angled  triangle. 

Imports,  (199),  goods  brought  into 
a  country. 

Improper  fraction,  (73),  one  whose 
numerator  equals  or  exceeds  its 
denominator. 

Indorse,  (214),  to  write  on  the  back 
of  a  business  paper. 

Indorsement,  (214,  215),  any  writ- 
ing on  the  back  of  a  business 
paper;  as  a  name  or  a  partial 
payment. 

Insolvent,  unable  to  pay  debts 
in  full. 

Inspection,  (64),  a  careful  exam- 
ination ;  obtaining  a  result  with- 
out working  the  example.  By 
inspection  I  find  that  2,  3,  5,  and 
11,  are  not  factors  of  223. 

Installment,(215),  a  part  payment. 

Insurance,  (194),  a  security  against 
loss  ;  the  value  put  upon  proper- 
ty to  be  paid  in  case  of  loss. 

Integer,  (72),  a  number  of  one  or 
more  units;  that  part  of  a  deci- 
mal at  the  left  of  the  decimal 
point. 

Interest,  (204),  money  paid  for  the 
use  of  money. 

Kilo-,  (127),  a  metric  prefix  mean- 
ing 1000. 


Least  common  denominator,  (76), 
the  smallest  denominator  com- 
mon to  two  or  more  fractions. 

Least  common  multiple,  (68),  the 
smallest  nnniber  that  will  con- 
tain each  of  several  numbers. 

Liabilities,  (178),  the  debts  of  a 
firm  or  individual. 

Link,  (126),  a  division  of  a  survey- 
or's chain,  7.92  inches  in  length. 

Linear  unit,  (128),  any  line  taken 
as  the  unit,  as  the  foot,  yard,  or 
m  eter. 

Liter,  (leeter),  (141),  the  unit  of 
metric  liquid  measure;  equal  to 
1.0567  liquid  quarts. 

Long  division,  (53),  the  method  of 
writing  the  work  in  division  in 
full. 

Longitude,  (150),  distance  in  de- 
grees east  or  west  of  the  meri- 
dian of  Greenwich,  Eng. 

Loss,  (185),  the  amount  the  cost  of 
an  article  exceeds  the  selling 
price. 

Lowest  terms,  (75),  when  the  nu- 
merator and  denominator  of  a 
fraction  contain  no  common 
factor. 

Market  value,  (201),  the  price  of 

stocks  in  the  market. 
Maturity  of  a  note,  draft,  or  bill, 

(214),  the  date  when  it  tiecomes 

due. 

Means,  (176),  the  second  and  third 
terms  of  a  projiortion. 

Measuring  unit,  (127),  a  unit  in 
which  the  quantity  measured  is 
expressed. 

Mensuration,  (246),  measuring 
and  calculating  the  contents  of 
surfaces  and  solids. 

Meter,  (127),  the  unit  of  length 
from  which  the  decimal  system 
of  weights  and  measures  is 
named,  equal  to  39.37  inches. 

Metric  system,  (127),  the  decimal 
system  of  weights  and  measures. 

Milli-,  (127),  a  metric  prefix  mean- 
in  o-  _  J of 

Miner's  inch,  (157),  flowing  water 
at  the  rate  of  10.4279  gallons  per 
minute. 


268 


CALIFORNIA   SERIES. 


Minuend,  (21),  the  number  in  Sub- 
traction from  which  another  is 
taken. 

Minus,  (21),  less,  or  diminished  by ; 
the  name  of  the  sign  of  subtrac- 
tion. 

Mixed  number,  (73),  a  whole  num- 
ber and  fraction  combined. 

Mortgage,  (226),  a  grant  of  proper- 
ty to  a  creditor  as  security  for 
the  payment  of  a  debt. 

Multiple  of  a  number,  (07),  a  num- 
ber that  contains  it  an  exact 
number  of  times. 

Multiplicand,  (34),  the  number  to 
be  multiplied. 

Multiplication,  (34),  the  process  of 
finding  a  number  of  times  a 
given  number. 

Multiplier,  (34),  the  number  by 
which  we  multiply. 

Myria-,  (127),  a  metric  prefix 
meaning  10000. 

Negotiable,  (214),  can  be  transfer- 
red to  another  part}^,  as  a  note 
or  draft. 

Net  proceeds,  the  sum  remaining 
from  a  sale  after  the  payment 
of  all  expenses. 

Net  weight,  (200),  the  weight  of 
packed  goods,  not  including  the 
weight  of  cases,  or  packing  ma- 
terial. 

Notation,  (5),  writing  numbers. 

Note,  (214),  a  written  promise  to 
pay  money. 

Number,  (5),  one  or  more  units. 

Numeration,  (9),  reading  of  num- 
bers written  decimally. 

Numerator,  ( 72),  the  number  above 
the  line  in  a  fraction;  corre- 
sponds to  the  divisor  in  division. 

Odd  number,  not  containing  two 
as  a  factor. 

Oral,  spoken. 

Order,  (224),  a  written  direction  to 
one  person  to  pay  money  to 
another. 

Parallel,   (240),  having  the   same 

direction. 
Parallelogram,    (240),    a    4-sided 

figure  whose  opposite  sides  are 

parallel. 


Partial  payment,  (214),  payment 
of  a  part  of  a  note. 

Partners,  (178),  associates  in  busi- 
ness. 

Partnership,  (178),  an  association 
of  persons  to  carry  on  business 
together. 

Par  value,  (201),  nominal  of  face 

value. 
Payee,  (229),  one  in  whose  favor  a 

draft  or  check  is  drawn. 

Payer,  (229),  the  drawee  of  a  draft 
or  check. 

Per,  by. 

Per  cent,  (181),  hundredths;  liter- 
ally, by  the  hundred. 

Percentage,  (181),  a  number  ob- 
tained by  taking  a  per  cent  of 
another. 

Perch  of  masonry,  (138,  155),  10^ 
or  24|  cubic  feet,  according  to 
custom. 

Perimeter,  (247),  the  boundary 
line  of  a  polygon. 

Perpendicular,  (246),  at  right  an- 
gles with;  vertical. 

Personal  property  ,(198),movables, 
including  money  and  stock. 

Plane,  a  surface  straight  in  all  di- 
rections. 

Plus,  and,  or  added  to,  (14),  name 
of  the  sign  of  Addition. 

Poles  of  the  earth,  points  of  the 
surface  which  have  no  motion 
in  the  daily  revolution. 

Policy,  (194),  the  written  contract 
in  insurance. 

Poll  tax,  (198),  a  tax  assessed 
equally  upon  men  without  re- 
gard to  property. 

Polygon,  (246),  a  plane  surface 
bounded  by  straight  lines. 

Pound,  (157),  the  unit  of  English 
money,  .$4.86. 

Power,  (63),  the  product  of.a  num- 
ber repeated  as  a  factor. 

Premium,  (194),  the  percentage 
]iaid  for  a  policy  of  insurance; 
(202),  the  rate  of  the  market 
above  par  value  of  stocks. 

Present  worth,  (212),  the  present 
vahie  of  a  debt  due  at  a  future 
time. 


ARITHMETIC. 


269 


Prime  factor,  (63),  one  not  made 
np  of  other  factors. 

Prime  number,  (63),  one  not  made 
up  of  factors. 

Principal,  (204),  money  lent  at  in- 
terest. 

Prism,  (252),  a  solid  whose  side 
faces  are  parallelograms  and 
whose  ends  are  equal  parallel 
polygons. 

Problem,  something  to  be  done. 

Proceeds,  (190),  sum  left  after  tak- 
ing out  a  discount  or  commis- 
sion. 

Product,  (34),  the  result  of  multi- 
plying one  number  hy  another. 

Pront,  (185),  gain. 

Proof,  (18),  a  test  of  correctness  of 

work;  no  arithmetical  proof  is 

a  perfect  test. 
Proper  fraction,  (73),  one  whose 

numerator  is  smaller  than    its 

denominator. 
Proportion,  (176),  two  equal  ratios. 
Pyramid,  (252),  a  solid  of  plane 

faces  whose  base  is  a  polygon 

and  summit  a  point. 
Quadrant,  (146),  the  fourth  part 

of  a  circumference;  ninety   de- 
grees. 
Quadrilateral,  (246),  a  polygon  of 

four  sides. 
Quantity,  anything  that  can  be 

measured,  weighed,  or  counted. 

Quintal,  (142),  100  pounds,  by  the 
long  ton  table  112  pounds. 

Quotient,  (43),  the  result  of  di- 
viding one  number  by  another. 

Radius,  (146),  distance  from  the 
center  to  the  circumference  of  a 
circle. 

Rate  of  interest  or  discount,  (181), 
per  cent  for  a  given  time. 

Ratio,  (176),  the  indicated  division 
of  one  number  by  another  of 
the  same  kind. 

Real  estate,  (198),  lands  and 
houses,  immovable  property. 

Receipt,  (119),  a  written  acknowl- 
edgment of  something  received. 

Rectangle,  (128),  a  polygon  of  four 
sides  and  four  square  corners; 
a  right  angled  parallelogram. 


Reduction,  (123),  changing  a  num- 
ber; (74),  or  fraction  in  name 
without  changing  value. 

Remainder,  (21),  number  left  after 
taking  away  one  number  from 
another. 

Remittance,  (193),  money  or  an 
order  for  money,  sent  to  a  dis- 
tant place. 

Right  angle,  (146),  a  square  cor- 
ner. 

Roman  notation,  (12),  writing 
numbers  by  capital  letters  of 
our  alphabet. 

Root  of  a  number,  (237),  one  of 
its  equal  factors. 

Rule,  a  direction  for  working 
problems. 

Section  of  land,  (129),  a  mile 
square. 

Security,  a  pledge  for  the  pay- 
ment of  a  debt. 

Share,  (201),  one  of  the  equal 
parts  into  which  the  capital  of  a 
company  or  corporation  is  di- 
vided. 

Short  division,  (50),  division  in 
which  the  result  only  is  w^ritten. 

Sight  draft,  (229),  one  payable  on 

presentation. 
Simple  number,  (122),  a  multiple 

of  a  single  unit;   expressed  in 

terms  of  a  single  unit.  , 
Slant  height,  (252),  the  shortest 

distance  down  the  side  of  a  cone 

or  pyramid. 
Solid,  (134),  a  body  having  length, 

breadth,  and  thickness. 
Solution,  the    process    of  work- 
ing a  problem,  also  the  answer 

obtained. 
Specific  duty,  (200),  a  tax  on  the 

measure,  number,  or  weight  of 

imported  goods. 
Sphere,  (252),  a  solid  body  having 

a  uniformly  curved  surface. 
Square,  (128),  a  rectangle  of  equal 

sides ;  the  product  of  two  equal 

factors. 
Square  root,  (237),  one  of  the  two 

equal  factors  of  a  number. 
Standard  time,  (154),  the  time  of 

the   meridians  of  75°,  90°,  105°, 

and  120°. 


270 


CALIFORNIA   SERIES. 


Statement,  (226),  a  tabular  ar- 
rangement of  assets  and  liabili- 
ties. 

Stock,  (178,  201),  the  capital  of  a 
firm  or  comi)anj\ 

Subtraction,  (21),  the  process  of 
taking  one  number  from  an- 
other. 

Subtrahend,  (21),  the  number  to 
be  taken  from  another. 

Sum,  (14),  the  result  obtained  by 
adding;  money. 

Surface,  (128),  that  which  has 
length  and  breadth ;  the  outside 
of  a  solid. 

Symbol,  a  letter  or  other  charac- 
ter used  for  a  member. 

Tangent,  (146),  an  indefinite 
straight  line  touching  a  curve. 

Tax,  (198),  a  sum  charged  by  the 
government  or  other  authorit}^ 
upon  property  or  person. 

Terms  of  a  fraction,  (72),  the  nu- 
merator and  denominator. 

Time  draft,  (229),  one  payable  a 
certain  specified  time  after  pre- 
sentation or  date. 

Trapezium,  (246),  an  irregular  four 
sided  polygon. 


Trapezoid,  (246),  a  trapezium  with 
two  opposite  sides  parallel. 

Triangle,  (246),  a  polygon  of  three 

sides. 

Troy  weight,  (143),  used  for  gold, 
silver,  precious  stones,  etc. 

True  discount,  (212),  the  differ- 
ence between  a  sum  due  at  a  fu- 
ture time  and  its  present  value. 

Unit,  (5),  a  single  thing;  a  collec- 
tion of  several  things  taken  as 
one. 

Value  of  a  fraction,  (72),  the  re- 
sult of  dividing  the  numerator 
by  the  denominator. 

Vara,  (154),  a  Spanish  measure  of 
length  equal  to  2.782  feet. 

Vertex,  (246),  the  point  opposite 
the  base  of  a  cone,  pyramid,  or 
triangle. 

Vertical,  (246),  at  right  angles  to 
the  horizon. 

Volume  of  a  solid,  the  product  of 
its  three  dimensions;  its  con- 
tents. 

Weight,  (142),  the  force  wdiich 
draws  a  body  downward. 

Width,  (246),  one  dimension  of  a 
surface  or  solid. 


ARITHMETIC, 


AE"SW 


271 


Exercise  22. 

i.  818. 
2.  393. 
^.  2756. 
4.  7931. 
J.  9448. 
6\  9135. 
7.  3099. 
<§.  7938. 
.9.  13872. 

10.  6752. 

ii.  11110. 

12.  26700. 
i^.  18701. 
14.  18287. 
i5.  13945. 
i<?.  10019. 
17.   11210. 

Exercise  23. 

1.  388. 
^.  286. 
^.  441. 

4.  421. 

5.  353. 

6.  392. 

7.  443. 
<^.  517. 
9.  417. 

i(?.  269. 

11.  346. 
i^.  454. 
iJ.  303. 
14.  418. 
i5.  168. 
16.  428. 
i7.  408. 
i5.  393. 
19.  350. 
^(?.  397. 
21.  335. 

Exercise  24. 

i.  7984. 

2.  6241. 
^.  7226. 

4.  8529. 

5.  4726. 

6.  5532. 
7.,  13507. 


<S.  2840. 

9.  11792. 

if.  16784. 

ii.  1066197. 

12.  1405655. 

13.  10674073. 
i^. ''5908144. 
15.  13100. 


29.  14029088. 

30.  136815. 

31.  658198. 
5^.  1731895. 
33.  584184. 
J^.  101102016. 
35.  20121022. 


16.   6862. 

Exercise  36 

i7.  970. 

1. 

44. 

18.  5528. 

2. 

221. 

i5.  8246. 

3. 

511. 

20.   5712. 

4. 

233. 

ii.  10042. 

5. 

401. 

^^.  5137. 

6. 

111. 

23.   23210. 

7. 

111. 

;.^-^.  6354. 

8. 

101. 

25.   1585323. 

9. 

102 

^^<?.  634650. 

t?7.  432724. 

Exercise  37 

1. 

196. 

Exercise  25, 

2. 

171. 

i.  1732. 

3. 

1476. 

2.  5036. 

4. 

70. 

^.  8082. 

5. 

531. 

^,.  30886. 

6. 

479. 

5.   221528. 

7. 

6899. 

6.  719733. 

8. 

199. 

7.  75008. 

9. 

528. 

5.  325466. 
9.  11094. 

Exercise  39 

10.  9105. 

1. 

188. 

ii.  25537. 

2. 

74. 

12.   8879. 

3. 

1326. 

iJ.  144224. 

4. 

3745. 

i^.  75359. 

5. 

162. 

15.   739557. 

6. 

7209. 

i6\  4139G. 

7. 

410. 

17.   1868. 

8. 

481. 

ic5.  38729. 

9. 

8591. 

i.9.  226139. 

10. 

504. 

20.   65584. 

11. 

3087. 

^i.  40596. 

12. 

6174. 

22.   76773. 

13. 

6190. 

^=?.  1104692. 

14. 

7092. 

^.^.  419302. 

15. 

2898. 

25.   707451. 

16. 

1728. 

^6.  422395. 

17. 

8730. 

27.  58849. 

18. 

100. 

^<?.  3344048. 

19. 

270. 

20.  370. 

21.  82. 

22.  241. 

23.  42(). 

24.  3852. 

25.  8970. 

26.  1158. 

27.  2211. 
h'.   718. ' 

29.  2904. 

30.  132. 

31.  2907. 

32.  bTiS. 

33.  8721. 

34.  7981. 

35.  497. 

36.  5229. 

37.  1492. 

38.  888. 

39.  1383. 

40.  22. 

Exercise  40. 

1.  3922. 

2.  269. 

3.  82. 

4.  3153, 

5.  953. 
6*.  161. 
7.  3959. 
5.  3493. 
9.  4573. 

if.  4982. 
ii.  1260. 
12.   780. 
iJ.  7198. 
14.   304. 
i5.  389. 
i6\  707. 
17.   1262. 
i<?.  1106. 
19.   190. 
fa  3131. 
fi.  70. 
22.  4248. 
;^^.  4113. 
24.   689. 
f  J.  3219. 
f6.  3299. 
27.  5601. 


272 


CALIFORNIA    SERIES. 


28.  1709. 

29.  493. 
SO.  344. 

31.  924. 

32.  943. 
5J.  2480. 
3/i..  681. 
55.  7168. 
56\  6626. 
S7.  4014. 
55.  5517. 
39.  8493. 
^0.  8267. 
^i.  352193. 
Ji2.  1884. 
^5.  19904. 
41,..  571010. 
^5.  515136. 
^.(?.  497707. 
Ji7.  539997. 
^^.  145204. 
J,9.  99182. 
50.  404884. 
5i.  698238. 
52.  356048. 
55.  3834791. 

54.  965149. 

55.  5004054. 
56\  397949. 
57.  19861. 
5<?.  518058. 
50.  3699962. 
60.  3490819. 

Exercise  41. 

1.  4191. 

2.  351. 

3.  792. 
^..  4120. 

5.  409. 

6.  3722. 

7.  6894. 

8.  85. 
5.  156. 

i6>.  1296. 

11.  4318. 

i^.  135. 

13.  80. 

i^.  2302. 

i5.  149. 

m.  1268. 

i7.  3161. 

18.  6487. 

i9.  1503. 

^a  2976. 

21.  350309. 

^^.  18020. 

23.  42290. 


^^.  394793. 

^5.  342190. 

26.  4190839. 

^.  417810. 

28.  498197. 

f.9.  4273. 

50.  3167. 

31.  851. 

5^.  7283. 

33.  34. 

5^..  205. 

55.  5521. 

36.  Tib. 

37.  4007. 

38.  6990. 
50.  370213. 
-^0.  102914. 

41.  3492601. 

42.  100248. 

Exercise  42. 

i.  390. 

^.  22. 

5.  3332. 

^.  153. 

5.  6886. 

^.  1. 

7.  405. 

8.  8595. 
0.  890. 

10.  810. 

ii.  3155. 

i^.  484. 

13.  7466. 

i^.  4372. 

15.  2827. 

i6\  63. 

i7.  110. 

18.  497. 

iO.  1031. 

20.  202. 

^i.  166. 

^^.  3520. 

23.  35. 

^^..  7074. 

25.  187. 

^e.  217. 

f7.  8783. 

28.  1078. 

^0.  998. 

30.  3343. 

5i.  672. 

5^.  7654. 

33.  4560. 

5.^.  3015. 

35.  251. 

56.  298. 

57.  685. 


38.  1219. 

50.  4. 

40.  364. 

^i.  3718. 

^^.  233. 

43.  7272. 

-^^..  385. 

45.  19. 

46\  8981. 

^.7.  1276. 

48.  1196. 

^5.  3541. 

50.  870. 

5i.  7852. 

5^.  4758. 

53.  3213. 

5^.  449. 

55.  496. 

56.  883. 

57.  1417. 

58.  86. 
50.  282. 
60.  3636. 
(5i.  151. 
6^.  7190. 
63.  303. 
6^.  101. 
65.  8899. 
66\  1194. 

67.  1114. 

68.  3459. 
60.  788. 

70.  7770. 

71.  4676. 
7^.  3131. 
73.  367. 
7^.  414. 

75.  801. 

76.  1335. 

Exercise  43. 

1.  3269. 

^.  601. 

5.  6868. 

-^.  7247. 

5.  300. 

6.  4925. 

7.  4903. 

8.  1213. 
0.  130. 

iO.  218. 

11.  383. 

i^.  197. 

13.  4171. 

i^.  2917. 

i5.  1266. 

16.  2. 

i7.  2475. 


24. 
25. 


18.  204. 

iO.  221. 

20.  2426. 

^i.  198. 

22.  368. 

2634. 

1497. 

3347. 

26.  3344. 

^7.  647. 

^<5.  4997. 

29.  4200. 

50.  299. 

31.  378. 

5^.  161. 

55.  5514. 

34.  4378. 

55.  5866. 

36.  1486. 

57.  3116. 

55.  3612. 

39.  183. 

^0.  155. 

41.  302913. 

^^.  645913. 

^5.  299400. 

44.  540987. 
^5.  550032. 
46.  889. 
^7.  5106. 

45.  346013. 

49.  375020. 

50.  170214. 

51.  4035600. 

52.  3691753. 

53.  499978. 

54.  5099931. 

55.  956940. 

56.  420042. 

57.  309007. 
55.  556295. 

Exercise  44. 

1.  3870. 

2.  6267. 
5.  4625. 
.^.  22. 

5.  88. 

6.  165. 

7.  1254. 
5.  4183. 
0.  2679. 

10.  17. 

ii.  566. 

12.  3002. 

i5.  3. 

i4-  2697. 

15.  3901. 


ARITHMETIC. 


273 


16.  79. 

17.  1136. 

18.  1488. 

19.  49f5. 

20.  3429. 
^i.  343000. 
f^.  34G513. 
2S.  3045. 
^^.  549143. 
25.   351119. 
£6'.  29007. 
^'.  4205814. 
28.  343847. 
^;7.  4599953. 

30.  4142991. 

31.  111035. 
J^.  247288. 
33.  2998. 
^4.  278. 
35.  295. 
^6\  12. 

-57.  2458. 
38.  3200. 
^^9.  650. 
40.  4279. 
^i.  7002. 
#.  313. 
43.  43600. 
.^4.  2156. 
45.  23901. 
^6.  514061. 
^7.  3643013. 
4S.  667330. 

Exercise  47. 

1.  3463  A. 

^.  1326  trees. 

3.  $230. 

.^.  66  mi. 

5.  365  days. 

6\  $3081. 

7.  5137  people. 

<?.  63  vr. 

9.  ISIo.,  231  mi. 
10.  39373  votes. 
ii.  9141  votes. 
,^  )  $485,  drew; 
^^-  t  $1890,  rem. 
(91145,  both; 

13.  <  58905,  more 

(     Chinese. 

14.  85  nut  trees. 

15.  $27. 

i6\  Wash.;  11  yr. 

( 1401 ■ 
17.  <  3052 

(5729 
i<?.  1782  A 


cen- 
tals. 


18— A 


3311  sheep. 
$210. 
618. 
$275. 

2404  mi.  to  C. 

3367  mi.   to 
N.  Y. 
S.  F.  to  0. 461 

mi.  further. 
19  1957  votes. 
13181  votes. 
7855  votes. 
$13. 
222. 
459! 
843. 
$1100. 

77  marbles. 
1519. 

$6100. 
581  pupils. 
308  girls. 
1790. 
11699902. 
(  18199  mi.; 

1  4983  mi. 
181  days. 
170  ct.,  one; 

120  ct.,  the 
other;  50  ct. 
more. 

$1487. 

$204. 

1480 

424  centals. 

2352  B.  C. 

14162  ft. 

482  mi. 

$145. 

$505. 

303  sh'p;  $873. 

$2450. 

$11300. 

1208  arrests. 

153  days. 

818  years. 

$1975. 

2  days. 
1848. 
30379  ft. 
115  ct. 

56  1.  trees. 

Exercise  58. 


19. 
20. 

22. 
23. 

24. 

25. 

'27. 
28. 
29. 
30. 
31. 
32. 
33. 
34. 
35. 
36. 
38. 
39. 
40. 
41. 
42. 
43. 

44. 
45. 
46. 


1.  1448. 

2.  1566. 

3.  1284. 


4.  369. 

5.  886. 

6.  3555. 

7.  24G3. 

8.  2440. 

9.  28488. 

10.  16804. 

11.  4608. 

12.  45055. 

13.  28088. 

14.  9966. 

15.  12696. 

Exercise  59. 

1.  125. 

2.  1673. 

3.  3924. 

4.  12792. 

5.  9444. 

6.  10284. 

7.  1636. 

8.  32200. 

9.  216. 

10.  5040. 

11.  2284. 

12.  894. 

13.  9171. 

14.  1416. 

15.  3780. 

16.  32064. 

17.  1530. 

18.  3339. 

19.  7104. 

20.  10008. 

21.  4476. 

22.  22955. 

23.  3699. 

24.  61383. 
^5.  87849. 
26.  20020. 
^7.  1978. 
28.  28068. 
^5.  25680. 
^0.  12609 
31.  3282. 
5^.  57672. 

33.  rrm. 

34.  46900. 

5J.  11750. 

36.  9093. 

^7.  4420. 

38.  15132. 

^9.  59994. 

Exercise  61. 

/.  50. 
^.  75. 


3.  100. 

4.  150. 

5.  175. 

6.  200. 

7.  225. 

8.  478. 
5.  717. 

i(?.  956. 
11.   1195. 
i^.  1434. 
13.   1912. 
i^.  2151, 
i5.  872. 
16.   1308. 
i7.  1744. 
18.   2180. 
i9.  2616. 
;?(?.  3052. 
21.   3488. 
^f .  6396. 
23.   9594. 
^4.  15990. 
;?5.  19188. 
26.   22386. 
^.  25584. 
28.  28782. 
^5.  14166. 
JO.  18888. 
31.   23610. 
J^.  28332. 
33.   33054. 
^4-  37776. 
.55.  42498. 
36.   6856. 
J7.  13712. 
38.   17140. 
59.  20568. 

40.  23996. 

41.  27424. 
4^.  30852. 

43.  818. 

44.  1227. 

45.  2045. 

46.  2454. 

47.  2863. 

48.  3272. 
45.  3681. 
J(?.  9200. 
51.   13800. 
5^.  18400. 

53.  23000. 

54.  27600. 
J5.  36800. 

56.  41400. 

57.  72. 

58.  108. 

59.  144. 
6(?.  180. 
61.  252. 


274 


CALIFORNIA   SERIES. 


62.  288. 

63.  324. 

64.  2016. 

65.  3024. 

66.  4032. 
(J7.  6048. 
68.  7056. 
65.  8034. 
ZO.  9072. 
71.  1142. 
7^.  1713. 
73.  2855. 
7.:^.  3426. 

75.  3997. 

76.  4568. 

77.  5139. 

78.  593. 

79.  1192. 
<§(?.  1490. 
81.  1788. 
<5^.  2086. 
83.  2384. 
,?^.  2382. 
55.  2038. 
86.  3057. 
,^.  4076. 
88.  5095. 
59.  6114. 
9(9.  7133. 
91.  8152. 
9^.  472. 
93.  708. 
9^.  944. 

95.  1180. 

96.  1652. 

97.  1888. 

98.  2124. 

99.  1512. 
iW.  2268. 
101.  3024. 
i9;?.  4536. 
103.  5292. 
i04.  6048. 
i6'5.  6804. 
106.   8016. 
i97.  12024. 
108.   16032. 
199.  20040. 
ii9.  24048. 
111.  28056. 
ii^.  36072. 
113.   2295. 
ii^.  3030. 
ii5.  3825. 
116.  4590. 
ii7.  5355. 
118.  6120. 
ii9.  6885. 
i^9.  954. 


121.  1431. 
i;?^.  1908. 
123.  2385. 
ii?^.  2862. 
i.^5.  3816. 
126.  4293. 
i^7.  1776. 
128.  2664. 
7^9.  3552. 
im  4440. 
131.  5328. 
15^.  6216. 
133.  7992. 
iJ^.  2224. 
iJ5.  3336. 
136.  4448. 
iJ7.  5560. 
138.  6672. 
iJ9.  7784. 
i^.9.  8896. 

Exercise  62. 

1.  1290. 

2.  15750. 
^.  30000. 
^,.  143400. 
5.  1570500. 
6\  2187000. 
7.  214000. 
5.  15022000. 
9.  333500. 

10.   750000. 
22.  540000. 

Exercise  63. 

1.  13675. 

^.  130733. 

3.  238492. 
^.  1749306. 
5.  2582934. 
(?.  1875116. 

7.  223723. 

8.  2516200. 

9.  19392. 
10.  551376. 
2i.  180225. 
i^.  1722951. 

13.  3143124. 

14.  23054382. 

15.  34040898. 

16.  24712452. 

17.  2948481. 

18.  33161400. 

19.  259524. 

20.  7266672. 

21.  202000. 

22.  1931120. 


23.  3522880. 

24.  25839840. 

25.  38153760. 

26.  27698240. 
^.  3304720. 

28.  37168000. 

29.  290880. 

30.  8144640. 

31.  167500. 

32.  1601300. 

33.  2921200. 

34.  21426600. 

35.  31637400. 

36.  22937600. 

37.  2740300. 

38.  30820000. 

39.  241200. 

40.  6753(]00. 

41.  58750. 

42.  561650. 

43.  1024600. 

44.  7515300. 

45.  11093700. 
4.6.   8055800. 
47.  931150. 
4.8.   10810000. 

49.  84600. 

50.  2368800. 

51.  15715. 

52.  724409. 

53.  1321516. 

54.  9393138. 

55.  14312382. 

56.  10390268. 

57.  1239;;79. 

58.  13942600. 

59.  109116. 

60.  3055248. 

61.  27625. 

62.  264095. 

63.  481780. 

64.  3533790. 

65.  5217810. 

66.  3787940. 

67.  451945. 

68.  5083000. 

69.  39780. 

70.  1113840. 
7i.  189150. 
7^.  1808274. 

73.  3298776. 

74.  24196068. 

75.  35726652. 

76.  25933248. 

77.  3094494. 

78.  34803600. 

79.  272376. 

80.  7626528. 

81.  249975. 


82.  2389761. 

83.  4359564. 

84.  31976802. 

85.  47215278. 

86.  34276572. 

87.  4089591. 

88.  45995400. 

89.  359934. 

90.  10078992. 

Exercise  65. 

1.  $925. 

2.  768  hr. 

3.  $176. 

4.  95040  ft. 

5.  552  mi. 

6.  9372  trees. 

7.  $1164625. 

8.  $72000. 

9.  22490  lb. 

10.  $900. 
ii.  $552. 

12.  $494. 
i5.  $6300. 

Exercise  74. 

i.  1233. 
^.  411. 

3.  4242. 

4.  91. 

5.  91. 

6.  31. 

7.  641. 

8.  532. 

9.  501. 
i9.  1001. 

11.  701. 
i^.  301. 

Exercise  76. 

1.  29|. 
^.  26. 
3.  19|. 
^.  12f. 
5.  161. 
5.  17. 

7.  21. 

8.  18f. 

9.  39*. 
10.  22|. 
i2.  88^. 
i^.  72. 

13.  19L 

14.  102. 

15.  lOU. 
i6.  214i. 
i7.  58^. 


ahithmetic. 


275 


18.  60. 

19.  64^ 

20.  758§. 

21.  1400|. 

22.  888|. 
^5.  949. 
^^.  lllf. 
^5.  1432. 
;?6.  813|. 
27.  1229. 
^.^.  H75. 
;g.9.  1151. 

30.  3401f. 
^2.  145402|. 
32.  6492|. 
5J.  11250. 
5^.  144701i 
35.  19575|. 
56.  9889. 
37.  2546C§. 
5<?.  7350. 
59.  57578|. 

Exercise  76. 

1.  218104. 

2.  109052. 
5.  87241f. 

r2701|. 
62315^. 
54526. 

7.  48467|. 

<?.  29218. 

9.   19478§. 
i6>.  14609. 
11.   11687i. 
i;?.  9739|. 
i5.  8348. 
14.   7304|. 
iJ.  45000. 
16.   30000. 
i7.  22500. 
i5.  18000. 
19.   15000. 
2(?.  128571. 
21.   10000. 
^^.  361753. 
£5.  241168§. 

24.  180876f. 

25.  120584f. 

26.  103358. 
^.  90438|. 
^c^.  80389|. 
29.  58726. 
56>.  39150|. 

31.  29363. 
5^.  23490a. 
55.  16778f 
34.   14681|. 


6. 


35.  13050S. 
.%\  44500i. 
37.  29667r 
55.  222501 
55.  17800^. 
40.  14833|. 
^/.  12714f. 

42.  U12^. 

43.  38200. 
^^.  19100. 
45.   15280. 
^(>.  12733a. 
47.   10914f. 
^5.  9550. 
^.9.  8488|. 
50.  4900. 
Ji.  3075. 
52.  2940. 
55.  2450. 
5^,.  2100. 
55.   1837|. 
J(?.  1633|. 
57.  259103. 
55.  172735J. 
5.9.  129551|. 

60.  1036411. 

61.  86367|. 

62.  74029a. 
65.  64775|. 

Exercise  78. 

1.  350  2  rem. 

2.  400. 

5.  159  16  rem. 
4-  50  1  rem. 

5.  fi^  16  rem. 

6.  233  12  rem. 

7.  ^6Vi  20  rem. 

8.  126  16  rem. 

9.  55  11  rem. 
29.  143  6  rem. 
i2.  275  2  rem. 
2^.  200. 

25.  94  36  rem. 
24.  ^5  1  rem. 

15.  107  16  rem. 

16.  140  2  rem. 

17.  160. 

i5.  75  46  rem. 

19.  20  1  rem. 

20.  85  46  rem. 
^2.  116  42  rem. 
f;?.  255  20  rem. 

23.  63  16  rem. 

24.  16  41  rem. 

25.  71  36  rem. 
;^6.  100  2  rem. 
^.  ii.^  20  rem. 


2:). 
30. 
31. 
32. 
33. 
34. 
35. 
36. 
37. 
38. 
39. 
40. 
41. 
42. 
43. 
44. 
45. 
46. 

48. 

49. 

50. 
51. 

52. 
53. 
54. 
55. 
56. 
57. 
58. 
59. 
60. 
61. 
62. 
63. 
64. 
65. 
66. 
67. 
68. 
69. 
70. 
71. 
72. 
73. 
74. 
75. 
76. 
77. 
78. 
79. 
SO. 
81. 
82. 
83. 
84. 
85. 
86. 


54  16  rem. 

14  21  rem. 
61  26  rem. 
87  42  rem. 
100. 

47  36  rem. 
12  41  rem. 
53  56  rem. 
77  72  rem. 

55  80  rem. 
42  16  rem. 

11  11  rem. 
47  66  rem. 
32  111  rem. 

12  58  rem. 
47. 

40  57  rem. 
119  9  rem. 
21  211  rem. 

5  58  rem. 
31 100  rem. 
^6  257  rem. 
79  109  rem. 
16  111  rem. 

6  58  rem. 
23  200  rem. 
20  57  rem. 
59  209  rem. 

13  11  rem. 
4  458  rem. 
18  400  rem. 
16  57  rem. 
47  309  rem. 

10  511  rem. 

4  58  rem. 

15  400  rem. 
13  257  rem. 
39  409  rem. 

9  211  rem. 

5  358  rem. 
13  300  rem. 

11  357  rem. 
34.  9  rem. 

8  111  rem. 
5  58  rem. 
11  600  rem. 

10  57  rem. 
£9  609  rem. 

7  211  rem. 
^  658  rem. 
10  400  rem. 

8  857  rem. 
£6  409  rem. 
218  208  rem. 
£?9  436  rem. 
45. 

361 1506  rem 
58  1452  rem. 
i.^  1208  rem 


57.  i9  1436  rem. 
55.  30. 

89.  241  506  rem. 

90.  39  452  rem. 
91. 109  208  rem. 
92.  14  2436  rem. 
95.  22  2000  rem. 
9.^.  259  3506  rem. 

95.  29  1452  rem. 

96.  87  1208  rem. 
97. 11  3436  rem. 
95. 18. 

99. 144  3506  rem. 
100.  23  2452  rem. 
i9i.  72  4208  rem. 
i9^.  9  4436  rem. 
103.  15. 

i9.^.  120  3506  rem. 
295.  i9  3452  rem. 

106.  62  2208  rem. 

107.  8  2436  rem. 

108. 12  6000  rem. 
i99. 103  2506  rem. 
2i9.  26  5452  rem. 

111.  54  4208  rem. 

112.  7  2436  rem. 
113. 11  2000  rem. 
ii^.  90  3506  rem. 
i25.  i^  5452  rem. 

116.  4-8  4208  rem. 

117.  6  4436  rem. 
118. 10. 

219.  59  3506  rem. 

120. 13  452  rem. 

Exercise  80. 


1. 
^, 
5, 
4. 
5. 
6. 
7, 
5, 
9. 

19. 

12. 

1^. 

15. 

14. 

15. 

16. 

17. 

18. 

19. 

20. 

21. 

22. 


110  21  rem. 
83  37  rem. 
67  12  rem. 
73  87  rem. 
65  76  rem. 
I|r7  34  rem. 
77  43  rem. 
70  1  rem. 
5^  33  rem. 
97  12  rem. 
.^2  39  rem. 
31  40  rem. 

25  33  rem. 
^  82  rem. 
24  82  rem. 
<^  10  rem. 
29  22  rem. 

26  40  rem. 
51  9  rem. 
36  46  rem. 
159  19  rem. 
120  40  rem. 


276 


CALIFORNIA   SEMIFS, 


23.  96  88  rem. 
^.  106  72  rem. 

25.  94  94  rem. 

26.  184  16  rem. 

27.  Ill  76  rem. 

28.  101  7  rem. 

29.  118  78  rem. 

30.  140  20  rem. 

31.  136  33  rem. 

32.  103  23  rem. 

33.  83  6  rem. 

34.  91^9  rem. 

35.  81  38  rem. 
^(5.  157  50  rem. 
^7.  5J  77  rem. 

38.  86  59  rem. 

39.  101  78  rem. 

40.  120  17  rem. 
^i.  403  32  rem. 
-^^.  ^6>5  19  rem. 

43.  245  44  rem. 

44.  270  49  rem. 

45.  240  49  rem. 
<^6'.  4.66  43  rem. 
^7.  .?>b-5  37  rem. 
4S.  256  1  rem. 

49.  301  30  rem. 

50.  355  24  rem. 
Ji.  7333  21  rem. 
J^.  55.:^;?  32  rem. 

53.  44J6  96  rem. 

54.  4056  80  rem. 

55.  440(>  14  rem. 
56".  8553  5  rem. 
J7.  5iP^  80  rem. 

58.  4630  38  rem. 

59.  5521  49  rem. 

60.  6510  38  rem. 
<5i.  990  26  reni. 
6;?.  7^.9  14  rem. 

63.  602  42  rem. 

64.  664  4  rem. 

65.  530  26  rem. 
66\  114s  41  rem. 
67.  6r>5  56  rem. 
6'c?.  628  32  rem. 
63.  733  55  rem. 
70.  872  12  rem. 
7i.  1525  25  rem. 
7^.  iiJJ  66  rem. 

73.  927  81  rem. 

74.  1022  64  rem. 

75.  909  9  rem. 
76\  1764  36  rem. 

77.  i6'7i  36  rem. 

78.  967  69  rem. 

79.  1139  19  rem. 

80.  1343  19  rem. 
<5i.  12262  48  rem. 


5^.  .9^75  56  rem. 

83.  7458  80  rem. 

84.  8221  58  rem. 

85.  7308  14  rem. 
<S'6\  I4.I86  20  rem. 
57.  <5'6'i5  14  rem. 

88.  7779  59  rem. 

89.  9158  24  rem. 
90. 10798  40  rem. 
S'i.  1990  42  rem. 
5^.  iJf^  62  rem. 
93. 1210  82  rem. 
94. 1334  60  rem. 
95. 1186  38  rem. 
ry6\  2302  50  rem. 
57.  i^5<?  20  rem. 
98. 1262  86  rem. 
99. 14.86  58  rem. 

100. 1753  1  rem. 
iC'/.  Ill  645  rem. 
i(^.v^  iJ.^  297  rem. 
103.  185  386  rem. 
104. 174  261  rem. 

105.  125  501  rem. 

106.  207  198  rem. 

107.  381  228  rem. 

108.  211  381  rem. 

109.  689  120  rem. 
iiO.  16  4057  rem. 
iii.  55  780  rem. 
112. 132  368  rem. 
113.  159  239  rem. 
114. 140  410  rem. 
ii5. 107  644  rem. 
ii(;.  276"  38  rem. 
117.  327  209  rem. 
118. 181  380  rem. 
119.  592  32  rem. 
2^^(9. 14  2074  rem. 
2^2.  i.b'  372  rem. 

122.  25  300  rem. 

123.  30  330  rem. 

124.  28  420  rem. 
i^5.  20  540  rem. 
/^6\  ^.^.  114  rem. 

127.  63  21  rem. 

128.  35. 

i^,9.  ii,5  123  rem. 

130.  2  4082  rem. 

131.  651  10  rem. 

132.  899  382  rem. 
iJJ.  1081 407  rem 
iJ^.  26*26^  46  rem. 
135.  731  658  rem. 
136. 1207  403  rem. 
137.  2224  14  rem. 
iJ<?.  1233  346  rem. 
iJ.9.  ^.6'27  13  rem. 
140.  97  3233  rem. 


141.  8  6348  rem.    I 
14:2.  23  1902  rem. 
143.  6  2036  rem. 
i^^.  7  2037  rem. 
145.  2  10818  rem. 
I4G.  IS  254  rem. 
147.  47  1926  rem. 
/.^cb^  12  4652  rem. 
2^.9.  2^  4654  rem. 
150.  4  22216  rem. 
151. 13  4358  rem. 
152.  36  513  rem. 
25^.  9  4401  rem. 
25-^.  22  374  rem. 
155.  3  17574  rem. 

Exercise  81. 


25  cows . 
31  da. 

11  mo. 
24  hr. 
93  A. 
35  mi. 
9!J0  A. 
346  chests. 

12  mo. 
10.  11  ponies. 


9. 


Exercise  82. 

2.  2250  yd. 

2.  21(3  T. 

3.  $21280. 

4.  4800  rd. 

5.  $341. 

6\  107  t\  T. 

7.  3525  ft). 

(^.  Blf  bales. 

.9.  14f  bags. 
2a  1800  eggs. 
11.  AOni  da. 
2^.  8760  hr. 
13.  $23800. 
2.^.  141  calves. 

15.  3  mi. 

16.  $1925. 

27.  $12624550. 
2c?.  gOOSi-^sV  sacks. 

19.  12iftf  centals. 

20.  $91.25. 
^2.  $200. 
^.    (  $4.20. 

"      (  42  loaves. 

23.  $1570. 

24.  Latter,  $5. 

25.  $49. 

26.  45  ct. 


28. 
23. 
30. 
31. 
32. 

33. 

34. 
35. 
36. 
37. 
38. 
39. 
40. 
41. 
42. 

43. 

44. 
45. 
46. 

47. 

43. 
49. 
50. 
51. 
52. 
53. 

54. 

55. 

56. 
57. 

58. 

59. 
60. 
61. 

62. 

63. 

64. 
65. 
66. 
67. 
68. 
69. 
70. 
71. 
72. 
73. 
74. 
75. 
76. 
77. 
78. 


j  1425  ct. 
t  150  ct. 
$5. 
400  ct. 

124  oranges. 
175  bbl. 
46375  K). 

f  288  mi.  a  da. 

(  12  mi.  anhr. 

1435  vd. 

$700.^ 

$84. 

714. 

65  mi. 

61  mi. 

10. 

25. 

$31835. 

j  $657. 

t  1314  da. 

8.302  f§  bales. 

Train,  480  mi. 

$1781. 

j  $6080. 

I  $960. 

60  da. 

55  sacks . 

125  cd. 
112  trees. 
144  da. 
375  boxes. 

(  63000  or'g's. 

t  5250  doz. 

$630. 

32  da. 

$100. 

j  15  watches. 

(  5  rings. 

32  mo. 

$2928. 

$2. 

$8820. 

f  $4166^«.. 

t  $200000. 

$14. 

$17. 

$104. 

$655. 

$2805. 

24  watches. 

$418. 

$50. 

$2. 

$2280. 

$240. 

14.553  cu.  in. 

$315. 

$180. 

$75. 


I 


ARITHMETIC. 


211 


79.  $4. 

8.  3. 

29.  660. 

iJ.  120  yd. 

80.  24  marbles. 

5.  15. 

50.  672. 

14.  340. 

81.  70  ct. 

i(?.  102. 

J2.  156. 

i5.  15. 

82.  45  vr. 

11.  18. 

32.  208. 

jQ  1  $300. 
-'^-  t  $1200. 

83.  225  %. 

i^.  3. 

<?5.  3450. 

84-  155  girls. 

13.  6. 

34.  99. 

,~    (  28  pupils. 
^^-  \  7  cards. 

85.  423  rd. 

i^.  16. 

J5.  312. 

<S'6'.  293  steps. 

i5.  9. 

J6\  600. 

18.  154  ct. 

87.  33  da. 

16.  47. 

37.  1344. 

v^  j  3  each. 
^^''  \  17  children 

88.  $15. 

i7.  31. 

J5.  612. 

IS.  71. 

39.  1530. 

Exercise  86. 

ifA  9. 

Exercise  98. 

1.  6. 

^^.  13. 
21.  29. 

Exercise  92. 

1-  W,  W- 

^.  11. 
3.  24. 

^^^  3. 
23.  3. 

i.  945. 

2.  1183. 

^.      9» ,    1 5 . 

3-  Hf  ^  W- 

^.  45. 

^-^..  3. 

^.  1014. 

-^-  i¥^,  -¥^. 

5.  9. 

^J.  2. 

4.  15708. 

<^-  W,  ^f^- 

g.  5. 

^6\  None. 

J.  12936. 

^     12  83    r,?.j 

7.  24. 

<?.  2583. 

"•      To    '    To  • 

7   is+i   aoji 

8.  None. 

Exercise  89. 

7.  1176. 

'^^      T    »    14  • 

5.  3. 

cS'.  1110. 

10.  4. 

1.  7. 

9.  105. 

.9.  W,  %V-- 

ii.  15. 

^.  21. 

2(?.  84. 

ia  5|i,  n4i. 

i^.  36. 

^.  41. 

ii.  80. 

11,    £|i     59» 

13.  35. 
i^.  9. 

4.  13. 

5.  53. 

12.  SCO. 
i^.  180. 

-'-'•       8    '    15  • 

15.  11. 

6.  31. 

14.  108. 

Exercise  101. 

i6\  20. 

iJ.  32. 

1.  1 

i7.  12. 

Exercise  90. 

i6\  567. 

IS.  15. 

17.  448. 

^-  #• 

iP.  30. 

i.  90. 

i<?.  132. 

•^-  t¥t' 

20.  13. 

2.  504. 

19.  23391. 

4-  t\. 

5.  f 

^i.  12. 

^.  40. 

^C.  33033. 

f;?.  7. 

4.  60. 

f  i.  609. 

^J.  7. 

J.  504. 

22.  32250. 

^^.  41. 

6.  300. 

^J.  38178. 

/.    #TT.^ 

25.  55. 

7.  120. 

24.  77499. 

'^.  1^.%. 

£6\  3. 

<§.  924. 

£(5.  5922. 

5-  It- 

£7.  13. 

9.  840. 

10.  #. 

28.  30. 

iO.  108. 

Exercise  94. 

/7    -V- 

^^9.  33. 

ii.  210. 

-'■-'■'    ITTI' 

30.  15. 

12.  60. 

1.  3  ft. 

^^.  fSf. 

5i.  12. 

ic?.  72. 

^.  8  ft. 

13.  H. 

5;?.   17. 

14.  240. 

3.  60  mi. 

U-  \%- 

33.  7. 

i5.  420. 

,   j  120  ft. 
'^-  1  7  lots. 

15.  ^\ 

^^.  2. 

itf.  750. 

-^^'    1  oIT' 

35.  14. 

17.  180. 

{ 80  mill. 

J6\  10. 

i5.  360. 

5.  <  400  rd. 

19.  4752. 

i  320  rd. 

18.  |. 

Exercise  88. 

i?(?.  462. 

J.  120  qt. 

i.9.  1. 

^i.  5544. 

C  105  t). 

^0.  ,v 

1.  135. 

22.  4200. 

7.  <  $10  B. 

^.  9. 

;?J.  630. 

($18  W. 

4>^     1 

3.  26. 

24.  840. 

<?.  840  ct. 

tCii.    2- 

^.  1001. 

^5.  105. 

.9.  120  ct. 

£5.  ;v 

5.  725. 

^^6\  570. 

10.  28  each. 

f^.  \i. 

6.  29. 

27.  108. 

11.  252  nuts. 

25.  f . 

7.  3. 

^<5.  390. 

12.  240  marbles. 

^f>'.  HI- 

278 


CALIFORNIA   SERIES. 


27.  \ 

28.  i 

29.  I 

30.  i 

31.  I 


34.  Jv 

36.1 
37.  f . 

38.1. 
39.  ^j. 

p.  ^. 
41.1. 

42.  i. 

43.  h. 


4e.  h- 
47.1 
4.8.  tV 
49.  if 

50.^. 

SI.  tV 
32.  i|. 
53.  f. 
54-  H- 

56.  |. 

57.  i. 

58.  f. 

59.  J. 
66».  l 
61.  f . 
6^.  if. 
63.  tV- 


67.  i. 

68.  |. 

6V).  *. 
70.  j%. 


Exercise  104. 

1      35      24     iSO 
■^*    47)  4T5»    4Tr* 

•^-   T:?-  >  "1T>  T7- 
?      18      4  0     J  -.  3 


5. 
6. 
7. 
(?. 
5. 

ii. 

12. 
13. 
14. 
15. 
16. 
17. 
18. 
19. 
20. 
21. 


20  .135  uiia 

40)     4C  '       4TI    • 


r.  375.  7 
5)  '9  >  9* 
IS       1       661 

TT)  IT)   IT  • 

_2  5       1_1 7_Q.      12 
ITJTT)    luo  >  TOTJ- 
tl       15.0     2  24 
TJ5T))    3oU)  35  0* 
15     10    lA 
4^)   48)  48' 


802      25  0     2  47 
■5B"TJ)   2S^(T)  2(TIi' 
5  0     2.004     45 
T"2)      72  "'7"^- 
49      3ii     36 
5B">   56)    5<I* 
33      14     21 
4^)   42)   4^- 
75  4  0     5  5  11     203 
^19')  "¥T9")1TT9 
80         64         49 

Tl¥)  IT'S)  Tr"?- 

1040      203       1 
""5B~)   "5  (T")   5^- 


Exercise  107. 


1. 
2. 
3. 

4- 
5. 

6. 
7. 
8. 
9. 

10.  2|^ 

12.  U. 

13. 

14. 

15. 

16. 

17. 

18. 

19. 

20. 

21. 

22. 


123 
•^30- 

119 

^24- 

5- 

1. 
117 


9  » 

It' 
If- 

41 
TTO- 

II 
5T5- 
31 

19 

04- 


25. 

26. 

27. 

28. 

29. 

30. 

31.  0, 

32. 


2- 
1  3 

H- 

15 
"58- 

1 
^¥' 

ft- 
3 


Exercise  109. 

1-  94^.. 
^.  22^f 
3.  92fi. 


^.  3if. 

5.  291^. 

6.  15i|. 
1871 

77tV 

20iS 


7. 

<5. 

5. 
10. 
11. 
12. 
13. 
14. 
15. 
16. 
17. 
18. 
19. 
20. 
21. 


35TJ- 

9723 


1259 

^25'0 

78IS 


1^ 


"^TT^" 

29f\: 

21tV 

loyj,. 


Exercise  111. 

1.  45.?^,  72^V 
^.  lOOyV,  24|i. 

^      1^7131      18104 

^.  140H,  13pft. 

5.  nm.  i6f i 

6.  84U,  62|4. 

7.  48HJ)  20^|. 

5.  239H,  m- 
q    1467     1735 

io.  I03ii  1031. 

11.  78H,  49/i.. 

r^     Q103     18113 

Exercise  114. 

4.  mn. 

k    59 

^-    U9- 

^        07 

7.  137^1  rd. 

8.  154if  lb. 

9.  65^^^  hr. 
10.  85if|  mi. 
ii.  57f§yd. 

12.  48^  ft. 

14.  162-1%%  cd. 
i5.  406^41  mi. 


63Hf  mi. 
286iH  A. 


21 

TTTI- 
63a  ft. 


i6 

17. 
18. 
19. 

20.  $124^. 

21.  95i^  hr. 

Exercise  116. 

1.  505J^,  801t'TT- 
^.   1207|,  281tV 


3.  639A,  209H- 
^.  1549|,  I56J4. 

5.  178,  229|. 

6.  mm,  828tV 

7.  644f ,  272t\. 
5.  31241,  i3H_i^. 
9.   258i%,  309i|. 

10.   1771f,  18  K)^. 
ii.  13401,  848/^. 

12.  173/^,  328^. 

Exercise  119. 

1.   $193|. 
f .  $49f. 
3.  $10480. 
^.  378  gal. 

5.  $1921. 

6.  39i^"mi. 

7.  22i|  hr. 
<?.  74f  pp. 
9.   $8746. 

iO.  440  yd. 

Exercise  122. 

1.  10. 
;?.  19J. 
3.  20§. 
^.  15|. 

5.  12. 

6.  6. 

7.  0. 

8.  16. 

9.  llj. 
10.  18. 

ii.  9i 

13.  18f. 

i^.  I5f. 

i5.  12. 
16.  8f 
i7.  3. 
18.  6. 
i9.  lOf. 
^0.  13f. 
21.  194. 
£;^.  24. 
23.  16f . 
f^.  18f. 
^5.  12|. 
^6'.  19|. 
^7.  225. 
28.  44|. 
^9.  40. 
50.  31^. 
31.  lOj. 
5^.  3§. 
33.  14^. 
5^.  45. 
J5.  51S. 


72' 


ARITHMETIC. 


279 


36.  25. 

S7.  30. 

38.  35^. 

39.  11§. 

40.  6|. 
^i.  401. 
^^.  44^. 
43.  51§. 
^^.  20|. 
45.  27|. 
^6*.  15. 
^7.  35. 
4S.  42i 

Exercise  125. 

2-  rV- . 
<?.  f. 

|.  j%. 
J.  3. 

(J.  4- 

^!*- 

0      2  35 
'^^     35T- 

10.  f. 

11.  l 

12.  iU- 

13.  |. 

14.  If. 

77    _9  o_ 

■^'  •     151' 

Exercise  126. 

I.  31i. 

^-  21-i\. 

^.  85tf . 

^.  67f. 

5.  611. 

6.  64|f 

7.  704 

8.  48A. 

9.  64^. 

10.  99s. 

ii.  25|. 
i^.  52*. 
13.  83M. 
i^.  62f 

15.  12n;i. 
i6\  92:7^. 
i7.  33i7i3. 
18.  280813. 
i.9.  llOSifi. 
W.  2001  H|. 
21.  2411^. 
£^.  5448^1-0. 
23.  1037|i. 


:^^.  2515H. 
£5.  276/^. 
^S.  11132^V- 
^.  39371^. 

28.  igiVifA- 

Exercise  128. 


424S  lui. 

$Gir;|. 

268|  ct. 
2305g  lb. 
f40i. 
$'37-jf^. 
14911  mi. 
264-iJ^  mi. 


Exercise  134. 

1   i^ 

■'■■    TK' 
&      5  5 

■*•  -sr' 
^    73 

^-  m- 

5.  2r\. 

6.  ^. 

O  ^83 

''•  TouTJ- 

9.  1^. 

iO.  l|f 

ii.  if. 
i^.  Iff. 
13.  1^. 

i6. 2e. 


Exercise  135. 


11  chairs. 
70f  da. 

8ft  yr. 
13  dresses. 
13  dresses. 
4  sons. 
27  div. 

8.  9  steps. 

9.  bl  da. 
10.  12i  cu.  ft. 

Exercise  136. 


2.  3; 

5.  5f. 

5.  5. 

^-  #3- 

7.  Hf 


<S.  If. 

P.  8. 

10.  17. 

ii.  2^. 

12.  5 
i.5.  i 

i5.  I 

16.  8 

^7.  § 

18.  h 

19.  i 

20.  * 


Exercise  138. 

1.  150. 
^.  176. 

3.  428. 

4.  405. 

5.  1290. 
6'.  600. 
7.  1440. 
<?.  413. 
5.  273. 

10.  1001. 

Exercise  140. 

i.  $45. 

2.  I. 

3.  625  sheep. 

4.  $1890. 

5.  $18450. 

6.  $8610. 
7    5 

<?.  $4i. 

5.  $66§. 

10.  $15360. 

ii.  $3840. 

ii.  ,\ 

13.  I,  I 

14.  240  A. 

15.  2000  sheep. 

16.  $3. 
77  7 

18.  $525. 
ia  |. 
20.  14300. 

Exercise  142. 

i.  6  vd. 

2.  29i  A.,  $1176S. 

3.  mh 
4. 3. 

5.  $3948|. 
g.  16  K). 

7.  $3150. 

5.  61  coats. 


9.  $2.12^\. 

ia  10  1b.' 

ii.  76  A. 

12.  l 

13.  $29^1. 

14.  19  fd. 
i5.  1760  yd. 
16.  $490^\. 
i7.  8  lots. 
18.  m  yd. 
i5.  20|  ct. 
20.  18^V  T. 


23.  171  cattle. 
( $1,302  A. 

24.  <  $1953  B. 
{ $1519  C. 


I  10  suits. 
I  2  vests. 


f5, 

26.  6|?-  da. 
£7.  ll  centals. 

28.  720  hogs. 

29.  $246|. 

30.  $907^3. 

?7       1_3 

J^.  $72^. 

55.  A,  by  Ij  mi. 

f  All  8  da. 
5^.<^17f  A,  24  B, 

35.  $80. 

36.  $f. 
57.  f  yd. 

55.  52  sheep. 
39.  4  children. 
.^0.  $76.^ 
41.  $15. 
.^^.  $140. 
,.   (  $432  A. 
^-  t  $1293  B. 
44-  $170')^. 
^.5.  122. 

46.  $78  on  all. 

47.  85^9^. 
C$630  A, 

.^.  <  $12G0  B. 

i  $945  C. 
49.  85  ct. 

f$30o0  A. 
oO.  <  $5100  B. 

i$7140C. 

51.  $144f. 

52.  $4036-^. 

55.  800  ce"ntals. 
( 1S|-  sum. 

54.  <  n  dif. 

( if  prod. 
55.  56  2). 


280 


CALIFORNIA   SERIES. 


56.  $3. 

57.  $200. 

58.  $li5i 

59.  ^. 

^^  J  $100  A. 
^^-  1  $200  B. 
6i.  1692^  yd. 
62.  9i  cwt. 
(55.  4,7^  T. 
64.  87(J6hr. 
(55.  $90i 

.  j  $14.25  rec'd. 

•  t  $1.50. 


66 


67 


50  mi.  to  S.  .1 


(  50 

t  483  "  to  L.  A 

{ $2.05  1st. 

68.  <  $(3.15  2d. 

i  $4.10  3d. 

69.  93|f  mi. 

70.  li|  mi. 

71.  11  bbl. 

72.  G  collars. 

73.  $99x^5. 
7^.  $10. 

75.  $3.50. 

76.  $31. 

~-y   j  2?-  mi.R. 
^^-  t3'mi.  W. 
75.  $228|. 
79.  9  boxes, 
o^  j  $l(ii  Fred. 
^^-  1  $8  Frank. 

Exercise  154. 

1.  1^03.52207. 
£.  9242.12079. 
S.  28347.257307. 

4.  7703.1697. 

5.  1603.178. 

6.  .34407. 

7.  4489.9514. 

8.  12515.2574. 

9.  300.244187. 

10.  67.28521. 

11.  688.1695. 

12.  27231.6101. 

13.  52.872. 

14.  85.05435. 

15.  4041.1615. 

16.  11.62. 

17.  83.668. 

18.  320.39. 

19.  927.155. 
m  32.1. 
;^i.  12.71. 
22.  9.6585. 

Exercise  155. 

1.  75.015. 


^.  772.0686. 

^.  857.03592. 

4.  857.13709. 

5.  452.13. 

6.  778.36. 

7.  12.929G. 
5.  164.3105. 
9.  27134.6793 

10.  81.0972. 

Exercise  158. 

1.  5346.0196. 

2.  9.3925. 

3.  935318334. 

4.  5.89849. 

5.  526.50598. 
6'.  9.8786558. 

7.  104437.086. 

8.  .683774. 

9.  9.3925. 

10.  .oiry.m. 

11.  16.391375. 

12.  .0098125. 

13.  .875875. 

14.  .01643375. 

15.  173.7375. 

16.  .0011375. 

17.  112397.62534. 

18.  11155.4977534 

19.  9.71418448. 

20.  .38159121. 

21.  53109.3(>631. 

22.  10330.50018. 

23.  267.5593924. 

24.  1024657.893262 

25.  67.28549. 

26.  6.6781049. 

27.  .00581528. 

28.  .000228435. 

29.  31.793285. 

30.  6.18423. 

31.  .1(]01714. 

32.  613.399157. 

33.  49.5443949. 

34.  140.1421021. 

35.  1201.0782784. 

36.  190182.24225. 

37.  3338.2649. 

38.  3386.480798. 

39.  .491239749. 

40.  .987987. 

41.  .929584929. 

42.  2.629439441. 

43.  22.5354304<)4. 

44.  3568.3258725. 

45.  (;3.197(>29. 

46.  69.16820758. 

47.  .00921693729. 


48.  .01853727. 

49.  114120.93327. 

50.  1403.93799. 

51.  4378283.6829. 

52.  5684788.293. 

53.  407281.42407. 

54.  65521.2759. 

55.  41198.0259. 

56.  14037.99. 

57.  .74717643. 

58.  .00919191. 

59.  28.6656461. 

60.  37.219637. 

61.  2.66656663. 

62.  .4289831. 

63.  .2697331. 

64.  .09191. 

Exercise  160. 

1.  43.5. 

2.  34. 

3.  .98. 

4.  .5184. 

5.  .512. 

6.  42.66«. 

19.  453. 

20.  414. 

Exercise  162. 

1.  3757. 

2.  6.25. 

3.  6556.55. 

4.  3.925. 

5.  350.35. 

6.  6.5735. 

7.  69495. 

8.  .455. 

9.  9022.5263 +  . 

10.  895.4884  +  . 

11.  .779  +  . 

12.  .0303  +  . 

13.  4263.263+. 

14.  829.263  +  . 

15.  21.477  +  . 

16.  82252.6526  +  . 

17.  .02162  +  . 

18.  .0311  +  . 

19.  .5241  +  . 

20.  83.0022  +  . 

21.  1.47 +  . 

22.  1.6089 +  . 

23.  .000214  +  . 

24.  .0004  +  . 

25.  81.294  +  . 

26.  1  +  . 

27.  3118.882 +  . 

28.  4049.574 +  . 


29.  290.128 +. 

30.  46.0742 +  . 

31.  29.3475+. 

32.  10. 

Exercise  168. 

1.  151.3575  bales. 

2.  299.25  K). 

3.  $3339. 

4.  .25. 

5.  $4.50. 

6.  81.12  A. 

7.  12  books. 

8.  74.25  mi. 

9.  11110.501     cu. 

in. 

10.  $407. 

11.  16.5  rd. 

12.  187.46  mi. 

13.  31.241  mi. 

14.  1415.425  A. 

15.  16  pr. 

16.  7276.5  cu.  in. 

17.  121  rd. 

18.  240.43+ turns. 

19.  $17.50. 

20.  21  cd. 

21.  30.76  A. 

22.  $1538. 

23.  $100. 
^^.  .21§. 
^5.  63  cd. 
26.  $477. 

^7.  64  times. 

28.  336.6  rails. 

29.  22.4482  in. 
50.  127.5  gal. 
31.  $164.40. 
5^.  127.4  mi. 

33.  11  hr. 

5^.  150.7968  sec. 
35.  25132.8  mi. 

Exercise  172. 

1.  $25.12i. 
^.  $26.49."' 
3.  $4.87|. 
.^.  $18. 

5.  $59.50. 

6.  $100. 

7.  $7.55. 

8.  $6.05. 
5.  $6,281. 


I 


i 


30. 


Exercise  174. 

15887  ft. 


/  15887  f 
t  5295.6{i 


,y<i- 


ARITHMETIC. 


281 


^^    f  no  in. 
^^-  t  S.Oogi  yd. 
.^  j  5315  ft. 
"^^^  t  1.0066+  mi. 
^^  j  768  in. 
'^'^-  \  64  ft. 
.,    (  462  in. 
'^'^-  1  38.5  ft. 
^-    (  63810  in. 
'^^^  i  1772.5  yd. 
g..    (  17160  ft. 
^^-  \  3.25  mi. 
^.v   j  10596  ft. 
'^'-  \  2.00G8+  mi. 

693  in. 

;.5  rd. 


..    (  83G0  yd 


r5  mi. 


,n  j  159831  ft. 

^^-  1  5327.75  vd. 

41.  1485  in.  " 

43.  ^§4  mi. 

44.  .83J  mi. 
^.5,  120  rd. 

46.  .1553  +  rd. 

47.  .2047  +  mi. 
^,y.  .723  — . 

4d.  .1324  +  mi. 

50.  .40'i5  +  rd. 

51.  /WV- 

5^.  HIi  mi. 

53.  i%. 

K  r.  4921       TYIT 

Exercise  177. 

9.  5  yd.  2  ft.  2.3 
in. 

10.  38  yd.  2  ft.  2.8 

in. 

11.  3.59  4-  mi. 

12.  199566.11  +  me. 

Exercise  178. 

^„    f  39204  sq.  in. 
^^-  \  4351.0  sq.  ft. 

13.  98027  .sq.  ft. 
^6>.  130380  sq.  ft. 
n.  9293318:^   sq. 

yd- 

22.  24^  sq.  ft. 

23.  685  sq.  ft.  166 

sq.  in. 

24.  28sq.rd.25sq. 

yd.  8  sq.  ft. 

25.  10  A.  13  sq. yd. 

2).  .^n3 157.46 +"; 


Exercise  179. 

10.   64000000  sq.  1. 
12.   128o9  sq.  ch. 

14.  64110625  sq.  1. 

15.  8  A.  4  sq.  ch.  4 

sq.  rd.  90  sq. 

16.  253A.7sq.ch. 

6  sq.  rd.  146 
sq.  1. 

17.  617  A.2sq.ch. 

2  sq.  rd. 
IS.  1   sq.  mi.  345 

A.  7  sq.  cli. 
10.  470A.8sq.ch 
20.  I. 

\  A.  lost. 

,yj;cultivat'd 

8  'ch. 

6.89  A. 


21 


Exercise  180. 

3.  4.3217.68  sq.m, 
5.  6477.732  ares. 

Exercise  181. 

1.  39|  yd. 

2.  Each  46|  yd. 
<^   j  25§  yd. cross 
'^-  \  25'yd.length 

4.  $64.58^ 

5.  .$103,121. 
6'.  $32,741: 

7.  12(:01isq.yd. 
S.  1447' sq.  yd. 
0.  110|  sq.  yd. 

10.  $34.75. 

11.  $9.50. 

12.  $30.25. 

13.  $8.19. 

14.  10.8  M. 

15.  I'^O;  bricks. 
10.  2210  sq.  ft. 
17.  $43.66§. 

IS.  60  tiles. 

Exercise  183. 


M 


2910  cu.  ft. 

108|  cu.  yd 
2.  15ff  cd. 
12.  6  cu.  ft. 
IG.  625536  cu.  in. 

17.  426829  cu.  in. 

18.  3 cu. yd. lieu. 

ft.  752  cu.  in. 
10.  29108  cu.  in. 
20.  44  cu.  yd. 


21.  1440  cu.  in. 

22.  4725  R). 

23.  762048  cu.  in. 

24.  1014  ft. 

25.  12  cu.  yd.  19 

cu.  ft. 
20.  llif  cd. 
27.  $26.41. 
2S.  9#y  cu.  yd. 
20.  2930§  cu.  yd. 

30.  .$3005.86. 

31.  .75  cu.  yd. 

32.  .125cu.'ft. 

33.  yV  cvi.  yd. 

34.  31992  cu.  in. 

35.  26  cu.  ft.  561. 6 

cu.  in. 

36.  48  cu.  ft. 

Exercise  184. 

3.  7.866  cd. 

4.  6.75G48  cd. 

Exercise  185. 

1.  $511.64. 

2.  57309  bricks. 

3.  $832.61. 

4.  240  perclies. 

5.  1280  perches. 

6.  870181  bricks. 
^  j  18§  ft. 

'■  \  $1.40. 

0.  1621ft. 
10.  $16,681. 
ii.  .$15.12. 

12.  540  ft. 
i.?.  $62.98. 

14.  8  ft. 
i5.  240  ft. 

16.  280  ft. 
i7.  (.00  ft. 

15.  3  0  ft. 
10.  384  ft. 
m  13J4ft. 
21  7121  ft. 

Exercise  186. 

13.  257  pt. 
i^..  036  qt. 
15.  252  pt. 
i(>.  12! '5  pt. 

17.  19  pt 
i5.  66  qt. 

19.  255  pt. 

20.  62  bbl.  11  gal. 
21  37  bbl.  29  gal. 

3  qt.  1  pt. 


22.  75  bbl.  23  gal. 

Iqt. 

23.  343  bbl.  13  gal. 

1  qt.  1  pt. 

24.  214  bbl.  13  gal. 

25.  128  bottles. 

26.  $27.05. 
ir7.  $59.06. 
28.  $40.80. 

^^  ]  3000  cans. 
'^'^-  \  $525.00. 
J6>.  ii  bbl. 
31  23  gal.  2  qt. 
Ipt. 

5.^.  .0238+  bbl. 

Exercise  187. 

4.  5605.611.  or  kg. 
r    \  1481.28  gal. 
"•  \  12332.32  lb. 

Exercise  189. 

39.  /^  lb. 

40.  .2025  t). 
^i.  191  pwt. 
43.  .375  R). 


^^. 


T. 


45.  .15025  T. 

46.  8  centals  33 

ft).  5g  OZ. 

47.  1  cental  50  !b. 
4.8.  If. 

5a  .875  T. 

Exercise  190. 

fllb.  loz.  18 
pwt.  15.912 
gr.  TroY. 
15  OZ.  125.412 
gr.  Av. 

4.  74620.837  li. 

5.  119.2  h. 

6.  150937.5  li. 

Exercise  191. 

.9.  106526". 
10.  232°  9'  25". 
11  16°  52'  30". 
1^-  T?5^(J  quadr't. 

13.  i-,. 

14.  .03  circumf er. 

15.  \. 

16.  .3763  +  . 


282 


CALIFORNIA   SERIES. 


17.  h 

18.  f  30". 

Exercise  192. 

10.  13  hr.  20  mill. 

10  sec. 

11.  7  mill.  45  sec. 

12.  $312. 

13.  18925  sec. 

14.  1037052  min. 

15.  97863853  sec. 

16.  1  yr.  317   da. 

11  hr.    49 
min.  39  sec. 

17.  33  da.  2  hr.  35 

min. 

18.  4  da.  22  hr.  42 

rain.  9  sec. 

19.  109  da.  9  hr. 

40  min. 

20.  99  da.  3  hr.  59 

mill.  43  sec. 

21.  530  min. 

22.  13  hr.  42  min. 

19  sec. 

23.  31  da.  16  hr. 

25  min. 

24.  202  hr. 

25.  304  da.  4  hr. 

26.  fiji  da. 

27.  i. 
■lhr.48min. 

4  da.  9  hr. 
2S.  \  137  da. 23  hr. 

16  min.  48 

sec. 
23.  H  week 

30.  211  da.  16  hr. 

48  min. 

31.  .125. 

32.  355  da.  21  hr. 

33.  45  da.  15  hr. 
34^.  I  mo. 

36.  -iVtj  week. 

37.  ^^  da. 

3S.  4  yr.  239  da. 
1  hr.  48  mill 
39.  .446  +  weelc. 
4.0.  .333  +. 
41.  .0125. 

Exercise  193. 

8.  36°  16'  30". 

9.  21°  4'. 

10.  48°  37'. 

11.  73°  18'. 


12.  62°  3'. 

i5.  102°  41'  30". 

14.  131°  26'  45". 

15.  1  hr.  57  min. 

17  sec. 

16.  1  lir.  4  min. 

19|-  sec. 

17.  3  hr.  57  min. 

5!)yV  !?ec. 
i5.  3hr.'l3  min. 
364  sec. 

19.  7  hr.  51  min. 

42  sec. 

20.  35  min.  ISif 

sec. 
^i.  29  mill.  32*  se. 

22.  11  hr.  19  min. 

48|  sec. 

23.  8  hr.  18  min. 

58 j7^  sec.p.M 

24.  6  hr.'30  min. 

9|-  sec.  A.  M. 
next  da. 

25.  1  hr.  51  min. 

50Y^5sec.A.M, 
next  da. 
f  168°  56'  W. 
^.   J  62°  41'  W. 
"^^  "i  81°  26'  W. 
[  159°  49'  E. 
27.  46°  28'  E. 
2S.  152°  20' 22"  E. 
20.  108°  56' 30"  E. 

30.  133°  54' 38"  W. 

31.  84°  41'  W. 

32.  3'5°  4'  W. 

33.  New  York. 

34.  72°  35'  74"  W. 
^5.  13   minr  364 

sec.  past  6 
A.  M.  same 
da. 

36.  64°  45'  3"  W. 

57.  60°  57'  W. 

Exercise  194. 

1.  1  mi.  125  rd.  3 

yd.  2  ft. 

2.  22  yd.  2  ft.  10 

in. 

3.  19  rd.  1  yd.  2 

ft.  2  in. 
.^.  74  mi.  96  rd. 
2  yd.  3  in. 

5.  80  rd.  4  yd.  3 

in. 

6.  46  mi.  252  rd. 

2  vd.  8  in. 


7.  103  mi.  41  rd. 

5  yd.  3  ill. 

8.  3  yd.  1ft.  4  in. 

9.  3  yd.  1  ft. 

10.  240  rd.  3  vd.  1 

ft.  6  in. 

11.  3  mi.  60  ch.  1 

rd.  23  1. 

12.  109  A.  154  sq. 

rd.3  sq.  yd. 

6  sq.  ft.  72 
sq.  in. 

13.  80sq.  rd.5sq. 

ft. 

14.  96  sq.  rd.   14 

sq.  yd.  1  sq. 
ft. 

15.  19cd.3cd.ft. 

13  cu.  ft. 

16.  281    cu.    vd 

22/^  cu.  ft. 
.,.  (49ga].lqt. 
^^-  1  $19:70. 

834  boxes. 
$10.44. 


19 


(8J 
t$] 


il.  5grosslldoz. 

3. 
22.  6353  sheets. 
^^    (  £23  16s. 
"^-  t  $115.67. 

24.  $3,174. 

25.  60  Tr  7   cwt. 

44  1). 
^5.  241  T.  6  cwt. 

51  R).  4  oz. 
^7.  11  S).  9  oz.  18 

pwt. 

Exercise  195. 

1.  47A.89sq.rd. 

2.  3  mi.  234  rd. 

5  yd.  6  in. 

3.  12A.7.26|sq. 

ch. 

4.  19  cu.  vd.  14 

cu.  ft.  1464 
cu.  in. 

5.  2gal.2qt.lpt. 
$14,444. 
8  gal.  1  qt. 

7.  i  cwt.  59  B). 

10  oz. 

8.  3.25  Troy  oz. 

9.  93  fi).  15"oz. 


6 


10.  2  oz.  7  pwt.  6 

gr. 

11.  llb.2oz.6dr. 

13  gr. 

12.  2  yrs.  2  mo.  1 

wk.  4  da.  4 
hr. 

13.  2  mo.  1  wk.  2 

da.  15  hr.  40 
min.  31  sec. 

14.  10  da. 

i5.  1  wk.  6  da.  5 
hr.  17  min. 
16.8  sec. 

16.  $322.18. 

17.  193rd.5#5-vc1- 

18.  345600  sec. 

19.  20  yr.  1  mo.  1 

wk.  1  da.  4 
hr.  35  min. 
37J  sec. 

20.  6  T.    7    cwt. 

18t%  K). 
^i.  43  sq.  yd. 

22.  £9    2s.  8d.'  3 

far. 

23.  21°  36'  40". 

24.  5  t).  3  oz.  10 

pwt. 

25.  10.^.  7jd. 
^6.  2070  sq.  ft. 

Exercise  196. 

i.  45  mi.  258  rd. 
2  yd.  2  ft.  3 
in. 

2.  6  mi.  79  ch.  3 

rd.  11  1. 

3.  10  A.  148  sq. 

rd.  4  sq.  j'd. 
2  sq.  ft.  6 
sq.  in. 

4.  1275cd.46cu. 

ft.  1214  cu. 
in. 

5.  136bbl.22gal. 

3qt. 

6.  63  lb.  2  oz.  4 

pwt. 

7.  1027  T.  12  cen- 

tals 56  lb.  14 
oz. 

8.  608  yr.  9  mo. 

1  wk.  2  da. 
10  hr.  14 
mill.  24  sec. 
.9.  70  rd.  4  yd.  4 
in, 


ARITHMETIC. 


283 


10.  $623.51. 

11.  273 mi.  105 rd. 

1  yd.  1  ft.  G 
in. 
('18bbl.26gal. 

12.  <         2  qt. 

($1187. 

13.  The  wine,  $992 

14.  56  A.  Ill  sq. 

rd.  5  sq.  yd. 
7  sq.  ft.'^Sij 
sq.  in. 

15.  4  oz.  9  pwt.  6 

gr- 
Exercise  197. 

1.  1  mi.  8  rd.  4 
yd.  1%  in. 

£.  11  ch.  1  rd. 
20§1. 

3.  18  sq.   rd.   16 

sq.  vd.  8  sq. 
ft.  1411  sq. 
in. 

4.  3cd.  59cu.  ft. 

1454^^cu.in. 

5.  1   bbl.   8   gal. 

l^  pt. 

6.  3  K).  9  oz.  11 

pwt.  23*  gr. 

7.  91 1).  9f  oz. 

5.  1  yr.  5  mo.  2 
wk.3da.  11 
hr.  59  min. 
14j2^  sec. 

9.  70tVV  te- 
i^.  6  lb.  4||-  oz. 
i2.  371   A.  6    sq. 

ch.  2  sq.rd. 

40|^  sq.  1.     I 
12.  3  rd.  2   yd.  2 

ft.  3Ulin.    I 
i,?.  162  culft.  432' 

cu.  in. 

14.  46  cups. 

15.  352  rails. 

Exercise  198. 

1.  8  mi.  203   rd. 

4  yd.  7  in. 

2.  1202  ra.  yd.  18 

cu.  ft. 

2.  Metric,1201cvi. 

yd. 

3.  570cwt.63i!b. 

4.  56^\  mi. 

4.  Jfe«rtc,90396.97 
meters. 


5.  5cu.ft.700cu.  38.  200  rd.  6  ft.  4 


13. 
14 
15. 
16. 

17. 
17. 
IS. 
18. 

ID. 

20. 
20. 
21. 

22. 
23. 
24. 
25. 
25. 
26. 

27. 

28. 
29. 
30. 


35. 
36. 
37. 


in. 
J/efric,cu.me 

.152686385. 
120§  sq.  ft. 
4014489600  sq. 

in. 
Metric^  hekt. 

259.1093. 
2t1t  cd. 
Metric^  steres, 

7.846544. 
6rd. 

73  spoons. 
6  yr.  10  mo.  1 

wk.  3  da.  12 

hr.  24  min. 

18  sec. 
$135. 
$400.95. 
$2,881. 
.125  mi. 
225.28  centals, 
$386.29. 
Metric,  same. 
2723if  mi. 
Metric,    met. 

587420.78. 
1  K).  3  oz.  8 

pwt.  21  gr. 
$574.42. 
i¥('<nc,$573.96. 
2680    cu.   yd. 

20  cu.  ft. 

6  qt.  U  pt. 
90.75  cu.  ft. 
48f  lots. 
$22.96. 
Metric,%'l.Vih. 

I  da.  1  hr.  43 
min.  30  sec. 

158  A.  31  sq. 

rd. 
$18.28. 
1120  times. 
22   sq.    yd.   4 

sq.  ±t.  82  sq. 

in. 
56  yd. 
$21.40. 
278i|  cu.  ft. 
138  sq.  rd.  27 

sq.  yd.  2  sq. 

ft.  99  sq.  in. 

7  hr.  57  min. 
18  sec.  A.  M. 

II  hr.  37  min. 
20  sec.  A.  M. 

.375. 


38.  Metric,1007.7Cj 

meters. 

39.  3|  ft. 

40.  $10.41. 

41.  60  yd. 

41.  Metric,    53.99 

meters. 

42.  23J  yd. 

43.  $1015. 

44.  $.91. 

,.    ($209.04. 
^^-  1  2SS  yd. 
46.  ^  yd". 
4:7.  57  rd.  9  ft.  104 
in. 

45.  A  cd. 
4d.  $12.25. 

50.  1 

(  Lots  45x145 

51.  <         ft. 

(  Gain  $1300. 

Exercise  199. 

22.  256  K). 

23.  39  books. 
$54,621. 
64  1b." 
$10.50. 
$81. 
152  1b. 
$732. 
$1230.60. 
45i  yd. 
14  chickens. 
$141. 

34.  512  bags. 

35.  28  tons. 

36.  $601. 

37.  $22.50. 

38.  $(;3. 

39.  43  lb. 

40.  20  sheep. 

41.  $765. 
$403. 

1143  rolls. 
$664. 
$605. 

46.  $1859.26. 

47.  $6283.75. 
4^.  $591.36. 
40.  $18345.55. 

50.  $307.90. 

51.  76  cwt. 

52.  $188.58. 

53.  $592.54. 

54.  1333J  B). 


25. 
26. 
27. 
28. 
29. 
30. 
31. 
32. 
33. 


42. 
43. 

44. 
45. 


55.  $27.67. 

56.  $248^1^. 

57.  $15.52". 

58.  $124. 

59.  $250. 

60.  50  eagles. 

61.  200  half  dol- 

lars. 

62.  100     quarter 

dollars. 

63.  500  dimes. 

64.  297.6  half  dol- 

lars. 

65.  3681^  pieces. 

Exercise  200. 


1. 
2. 
3. 
4. 
5. 
6. 


$70. 
7  days. 
$189. 
517  mi. 
105  T. 
27  A. 
/ .  40  chairs. 

8.  $25. 

9.  9  men. 

10.  24  men. 

11.  $142.80. 

12.  $12.50. 

13.  11§  da. 
$891. 
22  cows. 

n  yd. 

$562.50. 

18.  $360. 

19.  $72: 
90  horses. 
18  da. 
4000  lb. 
201  yd. 


14. 
15. 
16. 
17. 


20. 
21. 

22. 
23. 
24. 
25. 
26. 
27 
28. 


2hin. 


24  da. 

9  brooms. 
5  men. 
Ida. 
$56. 
$3.15. 

10  ft. 
10  ft. 
14  da. 
32  gal. 

35.  $238. 

36.  $148. 

37.  87idoz. 

38.  239  lb. 

39.  3000  sacks. 

40.  84  weeks. 

41.  21  sacks. 


30. 
31. 
32. 
33. 
34. 


284 


CALIFORNIA   SERIES. 


Exercise  201. 


'■{ 


A.  $1150. 

B.  $690. 
$245. 


0. 


10. 


■  \  B.  $392. 

1st  $44. 

2d  $52. 
,  j  1st  $4000. 
^-  I  2d  $2000. 

A.  $1612. 

B.  $2015. 

C.  $2418. 

G.  $45,  $60,  $90. 
fA.  $666§. 
I  B.  $1000. 

7.  {  C.$1200. 

I  D.  $1333^. 
LE.  $1800. 

8.  83J  ct.,  $1100. 
CA.  54T. 

{  B.  75  T. 
i  C.  120  T. 

A.  $131. 

B.  $393. 

C.  $262. 
I  A.  $1800. 

11.  <   B.  $600. 

(  C.  $1200. 
j^    (  1st  $3571. 
^"-  t  2d  $642!i. 
J.    ($171.60. 
^^-  t  $257.40. 

(432. 

14.  <  576. 

(720. 
(  A.  $990. 

15.  <  B.  $750. 

( C.  $1350. 

Exercise  209. 

1.  $2500. 

2.  $432. 

3.  $37.50. 
4-   20%. 
5.   $6(). 

6'.  $4500. 
7.  $1728. 

S-i%. 
0.   $4555. 
10.   $6. 

^^-  9*%. 
1£.  $52. 
iJ.  $(!73.30. 
i^.  $450. 
15.  $1094.70. 

io.  m%. 

17.  $10. 


i5.  $27. 
i.9.  16§%. 
iSO.   $625. 

Exercise  212. 

i.  $99. 

f  $250  board. 
I  $125  cloth- 

2.  i         ing. 

I  $450      inci- 
i,         dentals. 

3.  $300. 

4.  ^%;  15%. 
^.  I;  12J%. 

6'.  $5. 

7.  $40. 

5.  141%. 
.9.  $40000. 

i(?.  $900. 

Exercise  214. 

/.  $25. 
£.  $9.90. 

3.  124%. 

^.  $4.50. 

$96  loss. 
$3744  8.  P. 

0.  20%. 

7.  331  %. 

S.  $1.20. 

£^.  $1.50. 
iO.  331  %. 
ii.  12%. 
if.  $1.37|. 
7.   J  $49  0. 
^'^-  t  .$42  S.  P. 

14.  20  ct. 

15.  8h  ct. 
i6\  11%. 
17.  80  ct. 

ici^.  11^%  loss. 
10.  2H%  gain. 
m.  2^%  loss, 
fi.  90    ct.,  $1.14, 
$1.32. 

22.  331%. 

23.  10  ct. 
f^,.  $15.13. 
25.  $37. 
^6.  $82.50. 


^^  i  $14.40 

<i:l%  Ic 
$1,944. 


^o   i  $1.94 
29.  $1.50. 


15 


Exercise  216. 

^    (  $112.50  loss. 
^-  1  $1387.50  !S.  P. 
,,    (  $20000  cost. 
"•  \  $20500  S.  P. 
o    (  $131  loss. 
^-  )  $13331  cost. 
,   I  3f  %  loss. 
^-  t  $1925  S.  P. 
.    ($85.25  gain. 
'^-  \  $1170.258.  P. 

6.  28tV%  gain. 
^  (.$2100  cost. 
^-  1  $2247  8.  P. 
o  j  $2502  cost. 
^-  \  .$417  loss. 
9.  20%  loss. 

.^    ($1,231  loss. 
^^-  t  $19,731  cost. 
;.    ($1350  cost. 
^^-  1  $1,395  8.  P. 

12.  $1690  8.  P. 

13.  $750  cost. 
i.^.  $654/v  cost. 

'  $138  loss. 

$782  S.  P. 
.p  j  $6300  cost. 
^^-  \  $5460  8.  P. 
17.  61%  gain. 
IS.  14f  %  gain. 

.$980  gain. 

"^980  8.  P. 
^^    ($51  cost. 
'^^-  \  $76.50  8.  P. 
21.  P:'§%rate. 
^j,    (  $3  loss. 
^"-  1  $12.  8.  P. 
23.  $18  cost, 
f^.  $1000  cost. 

Exercise  218. 

y    ($21  com. 
^-  1  $679  net. 

^.  !<■§%. 

^.  $566.50. 

^    I  $3000  value. 

^-  \  $120  com. 

5.  $4.99J-  ($5.00.) 

6'.  30  ct. 

7.  $315. 

5.  300  bales. 
$2905. 
9g^JjC.  per  lb. 

10.  800  vol. 

11.  $2920. 

;^   (  $80  com. 
^^-  ]  $.3420  net. 
i^.  21%. 


19   I  '^^ 
^^-  t  $7' 


9. 


21. 


J900( 


u-  34%. 

i5.  $4100. 
16.  141%. 
..V    ($262.50  com. 
^'  •  \  $.3937.50  val. 
IS.  $2100. 
i.9.  $1.59g. 
f(>.  79500  t). 
9000  !b. 

ct.  perft, 
;.^f.  1000  R). 
f  J.  $432. 
^ ,    (  $.506.25. 
"■^-  t  23750  lb. 
25.  2%. 

Exercise  220. 

.  (  2*%  rate. 
1  $6435  pro. 
2.   $8000  cost. 
^  j  $5463.41  C. 
'^^  1  $136.59  com. 
,  (  $900  8.  P. 
^-  \  $886.50  pro. 
r    (  $174.64  com. 
^-  \  $8557.36  pro. 

6.  21%  rate. 

7.  3%  rate. 
S.  6%  rate. 
9.  6§%  rate. 

.,.  (  $2140  cost. 

^^-  t  $42.80  com. 

ii.  11%  rate. 

12.  $1756  8.  P. 

13.  2.8%  rate, 
i^.  1%  rate. 
15.  $1188  cost. 
10.  $5  com. 

Exercise  222. 


(  $44  prem. 
\  $5456  loss. 
$4000. 

f%. 

$1980. 

$3750. 
0.  2%. 

.V    (  $100  prem. 
'    \  $7500  val. 

1|%. 
$6756. 

$6.34. 

u%. 

First. 

$2184. 

$58.32. 
100/ 

is%- 


9. 
10. 
11. 
12. 
13. 
14. 
15. 


ARITHMETIC. 


285 


16.  $3168. 

17.  $500. 

18.  $5.25. 

19.  $2116.80. 

20.  111%. 

Exercise  225. 


$30,50. 
$2900. 
$40.50. 
45  ct. 
$500000. 
j  $1000000. 
t  3  mills. 


^-  1  $7. 
9.  U%. 

10.  $86. 

11.  $873. 
A'.  $42.86§. 
i^.  $!^00000 
14.  12%. 

Exercise  227. 

1.  $43. 
;^.  $375. 

^.  $221.76. 

4.  $4448.30. 

5.  $813.50. 

6.  $3806. 

7.  $1084.50. 
5'.  $3. 

9.  $750. 

if.  $320. 

11.  $54. 

i^.  $18.80. 

iJ.  $18. 

Exercise  228. 

2.  $1022.50. 

,   j  80  sh. 

^-  t  $8160. 
.    (  80  sh. 
^-  I  $4800. 
6?.  4f|%. 
7.  F.N.-^-f%. 
o    (  160  sh. 
•^^  1  $13120  cost. 
9.  ^' early  214%. 

10.  Each  5%. 

11.  16  sh. 

12.  $376.87*. 

13.  11%.      - 
i^.  $251.25. 


16 


15.  40  sh. 
(  95  sh. 
\  $6317.50. 
Or.  Nav. 
12  sh. 
10  sh. 


Exercise  231. 


$11.91. 
$191.41. 


1$ 

1$ 
f  $57.20, 
\  $382.2( 
(  $146.19 

1.1 


$904.94. 

$142.03. 
1  $1166.28. 
(  $7.21. 
1  $591.71. 
j  $55.77. 
t  $781.61. 
(  $83.02. 
t  $470.97. 
(  $.17. 
t  $42.37. 


Exercise  232. 

1.  $6;  $7.50;  $9. 

2.  $43.54 ;  $60.95 

$(i9.66. 

3.  $57.46. 

4.  $94.07. 

5.  $109.69. 

6.  $59.48. 

7.  $83.77. 

8.  $11.23. 

9.  $14.65. 

10.  $16.92. 

11.  $180.24. 
i:^.  $154. 

Exercise  238. 

1.  4  yr.  6  mo. 
^    f  1  vr.  4  mo. 
^-  t  1  vr.  2  mo. 
3.  $500. 
^.  8%. 
^.  6§%. 

6.  $240. 

7.  $679.61. 
<?.  Umo. 
9.  $392.16. 

10.  $11.36. 

11.  Offers  equal. 

12.  1  yr.  6  mo.  20 

(ia. 

13.  $705.59. 


A'.  $147.06. 
iJ.  12^%. 
^6\6%. 
17.  $4.32. 
i,^.  $20000. 
19.  $760. 
^a  11  mo. 

21.  3"yr.  3  mo.  18 

da. 

22.  6%. 
^5.  $218.75. 
24.   $6250. 
^5.  $484. 

26.   Jan.  2,  1882. 
^7.  3  yr.  1  mo.  (5 
da. 

28.  $885.75. 

29.  $1.92. 

30.  6  %. 
^/.  $750. 

32.  3  yr.  1  mo.  3 

da. 

33.  92  ct. 
^^.  $780. 

36.  $1970.50. 


Exercise  240. 

1.   $577.38. 


3.  $576.67. 

4.  $285.99. 
J.  $603.49. 
e.  $200. 

7.  $1386.78. 
<?.  $325.08. 
9.  $209.45. 
i6>.  $228.95. 

Exercise  242. 

1.   $262.48. 
£.  $39.71. 
3.   $15.80. 
>^.  $13.75. 

5.  $70.23. 
6'.  $2().n0. 
7.  $73.33. 
<^.  $12.56. 
9.   $116.86. 

if.  $112.58. 

Exercise  243. 

1.  $690.67. 

2.  $840.93. 
^.  $1426.88. 
^.  $908.46. 
5.  $247.50. 


6.  $1835.82. 

7.  $519.44. 

,?.  $297.83. 

5.  $1141.73. 

10.  $1350.56. 

11.  $738.31. 

Exercise  244. 

i.  $256.50. 

^-  28|  %. 

^.  $3.75  more. 

4.  $1000. 

5.  $190. 

6.  $596.11. 

7.  $80. 

5.  $363.80. 

9.  $703.25. 

if.  $4.45. 

ii.  $2.40. 

12.  $571.20. 

Exercise  245. 

i.  $5008.33. 

^.  $577.10. 

.^.  $1565.52. 

6.  $4500. 

7.  $736.56. 

8.l7o' 

9.  $769.04. 

if.  *%  prem. 

11.  $186. 

i^.  875  francs. 

Exercise  248. 

1.  7  mo.  5  da. 

;g.  3  mo.  17  da. 

0(4  mo.  14  da. 
'^-  \  Aug.  22. 

4.  18  da. 

J.  June  2.3. 

6.  5  mo. 

7.  $330. 

5.  20  mo. 

9.  June  14, 1886 

if.  Oct.  15,  1884. 

Exercise  249. 

i.  $3.10. 

2.  94*  ct. 

3.  Gj^j  ct. 

i  9fl%. 

5.  291  ct. 

6.  10  ct. 

7.  45  ct.  gain. 

Exercise  253. 

L  32  rd. 

286 


CALIFORNIA   SERIES. 


58  rows. 
1280  rd. 
99  ft. 
80x40  rd. 
64  rd. 
8.54  rd. 
101.2  rd. 
A's,  $90. 

10.  $420. 

11.  64  in.  sq. 


2. 
S. 

4- 
6. 
6. 

7. 
8. 
9. 


Exercise  256. 

1.  21  in. 

2.  61.3+  in. 
S.  8  cu.  ft. 

4.  67.64  gal. 

5.  1176  sq.  in. 

6.  3.17  ft. 

7.  13824  cu.  in. 

8.  1.6  met. 

9.  79.875  cu.  met. 

Exercise  257. 

1.  14.14  ft. 

2.  13.23  ft. 

3.  17.35  ft. 

4.  50  ft. 

5.  8.54  ft. 

6.  22.8  ft. 

7.  20.95  ft. 

8.  64.62  ft. 

9.  241.4  ft. 
i^.  33.8  ft. 
11.  51.42  ft. 
i^.  83.67  ft. 
13.  56.57  rd. 
7^,.  20  rd. 
15.  15.59  in. 

j  10.(J  rd. 

•  1  112.5  sq.  rd. 

17.  45.08  ft. 

18.  92.45  mi. 

19.  3  ft.  .98  in. 
W.  20.59  ft. 


16 


Exercise  258. 

1.  180  sq.  rd. 

2.  390  sq.  rd. 
o  j  69.12  rd. 

''•  t380.13sq.rd. 

4.  4.55  ft. 

5.  18.46  sq.  rd. 
6\  7'sq.  ft.+ 

7.  10.4  sq.  yd. 

8.  15  rd. 

5.  1293  sq.  ft. 

10.  42  ft.  8  in. 

11.  m  sq.  ft. 
1£.  47|  in. 

13.  30  ft.;  20'  ft. 

14.  78.54  sq.  ft. 
iJ.  $89.68. 

16.  $1518.75. 

17.  $7208.85. 

18.  93  sq.  ft. 

19.  420+  times. 
100.399    rd. 

£0.  <      dia. 

251.6  rd.  cir. 

Exercise  259. 

1.  24  cu.  ft. 

2.  52  sq.  ft. 

3.  3.21i  sq.  ft. 

4.  179.07  sq.  in. 

5.  3.91  qt. 

6.  795.87^  cu.  in. 

7.  2.53  sq.  ft. 

8.  8.17  in. 

9.  3.1416 sq.ft. 

10.  73i+  times. 

11.  lcd.981-cu.ft, 
if.  6.14  bu. 

13.  1.85  sq.  yd. 

14.  18.16  in. 

15.  1.47  gal. 

16.  .58  qt. 

17.  11.55  in. 
ic?.  68.80  sq.  ft. 

19.  54400  cu.  ft. 

20.  21.38  cu.  ft. 


r  201032400  sq. 
^7  ]      mi. 
"^- I  268083200000 

l^     cu.  mi. 

22.  $1275.12. 

23.  $337.92. 

24.  3+  cu.  ft. 

Exercise  261. 

1.  $220.50. 

2.  $3.60. 

^.  $3720.47. 
,  j  $139.71  com. 
^-  1  56312  lb. 
5.  $20. 

f  104.48  me- 
^  J  ters  wide. 
^-  j  313.4+    me- 

[     ters  long. 
7.  86if . 
<§.  $17.65. 
5.  27108.31  ft. 

10.  4.83  ares. 

11.  $42240. 
i;i  121ff%. 

i-i.  109%  nearly. 

14.  188.32  meters. 

15.  $1. 

..  (  $6035. 
-^^-  t  $5793.60. 
17.  $600. 
ii?.  15  men. 

19.  30^  sq.  rd. 

20.  174  +  meters. 
;?i.  259  +  hektares 

22.  $4001.40. 

23.  502.25. 

24.  .2169+. 
^5  f  210  A. 
'^^-  t  330  B. 

^^    f  41.62  sq. met. 
"^-  t  25.25  cu.met. 
27.  56  da. 
^5.  64  A.;  72  B. 

23398  liters. 

23396    kilo- 
grams. 


r60A.144sq. 
rd.  A. 
77  A.  98  sq. 

30.  rd.  B. 

i  103    A.   74§ 
sq.  rd.  C. 
206  A.  149J 
sq.  rd.  D. 

31.  3  316  ft. 

32.  194630.4  kilo- 

grams. 

33.  93.23  hhd. 
g,  f  4s.  1.38d. 
'^4-  \  5.376  fr. 

35.  4.6  meters. 

36.  $1031.85. 
oy   j  $600. 

^•184%. 
38.  187.2  steres. 
.g    (  $10.31. 
^^-  t  $42.69. 

f  $1600  A, 
^(^.  <  $2000  B. 

($1800  C. 
^i.  546t%  ft. 
^f.  $5.25. 
.^.5.  198  ft. 
.^^.  18.29  sq.  me- 
ters. 

(36. 

45.  <  3,  7,  21. 

i  2-',  33, 17. 

46.  $3750. 
.^7.  84. 

48.  51cd.  77cu.ft, 

49.  $54. 

50.  4. 

5i!  1.556.10. 
5f.  83957.8    kUo- 
grams. 

53.  $761.42. 

54.  $538.85. 

55.  5y\  min.  past 

1  o'clock. 

5(7.  4y\   min.   be- 
fore 1  o'cl'k. 

57.  $1199.08. 


ARITHMETIC. 


28: 


I^^DEX. 


Accounts,  221;  cash,  221;  personal, 
224;  barley  field,  225;  dairy,  225; 
bank,  227. 

Addition,  14;  of  several  columns, 
20;  of  two  columns,  20;  practical 
work,  28;  of  common  fractions, 
77;  of  decimal  fractions,  108;  of 
compound  numbers,  158. 

Analysis:  general,  172;  in  multipli- 
cation, 36;  in  division,  46;  frac- 
tional, 93. 

Average,  235;  of  payments,  233. 

Balance  Sheet,  226. 

Bills,  119. 

Brick  Work,  137. 

Brokerage,  190. 

Cancellation,  86. 

Carpeting  Rooms,  132. 

Cash  Account,  221. 

Check,  form  of,  227. 

Circle:  area,  250;  parts  of,  146. 

Commission,  189. 

Compound  Interest,  216. 

Cone,  252;  frustum  of,  252. 

Cube,  134,241;  root,  241. 

Customs,  200. 

Cylinder,  252. 

Decimal  System,  5;  notation,  5; 
fractions,  102;  point,  5.  i 

Discount,  219;  in  stocks,  202;  true,   i 
212;  commercial,  218.  ; 

Division,  43;  short,  50;  long,  53;  ' 
general  principles  of,  55;  practi- 
cal work  in,  5^ ;  of  common  frac- 
tions, 87;  of  decimal  fractions, 
110;  short  methods  in,  118;  of 
compound  numbers,  162. 

Draft,  form. of,  228. 

Duties,  199. 

Exchange,  228;  by  postal  order, 
231;  by  check,  231. 

Factors,  63;  prime,  63;  special  di- 
rections for  finding,  63 ;  greatest 
common,  65 ;  cancellation  of,  86. 


Fractions,  72;  terms  of,  72;  im- 
proper, 73;  lowest  terms  of,  75; 
common  denominator,  76;  addi- 
tion of,  77;  subtraction  of,  77; 
practical  w^ork  in  addition  and 
subtraction  of,  79;  multiplica- 
tion of,  82;  cancellation,  86;  di- 
vision of,  87;  inverting  the 
divisor,  89;  complex,  91;  what 
fraction  one  number  is  of  an- 
other, 91;  finding  the  whole 
when  a  part  is  given,  92;  practi- 
cal work  in  analysis,  93;  oral 
review,  94;  written  review,  97. 
Decimal,  102;  United  States 
money,  105;  changing  from  com- 
mon to  decimal,  103;  circulating 
decimal,  107;  addition  and  sub- 
traction of,  108;  multiplication 
of,  109;  division  of,  110;  con- 
tracted multiplication  of,  111; 
contracted  division  of,  112;  prac- 
tical work,  113. 

General  Analysis,  172. 

Greatest  Common  Factor,  65. 

Insurance,  194. 

Interest,  204;  six  per  cent  meth- 
od, 20:5;  exact,  209;  ptoblems  in, 
210;  compound,  216. 

Least  Common  Multiple,  68. 

Longitude  and  Time,  150. 

Measures,  122;  long,  122, 154 ;  sur- 
veyor's long,  126;  metric  long, 
127 ;  surface,  128, 155 ;  surveyor's 
surface,  130;  metric  surface,  131; 
cubic  or  solid,  134,  155;  metric 
solid,  137;  lumber,  138;  liquid, 
139,  155;  metric  dry  and  liquid, 
141;  circular,  146;  time,  147;  Cal- 
ifornia, 155,  157;  beer,  155;  dry, 
155. 

Mensuration,  246;  lines,  angles, 
and  surfaces,  246;  right  angle  tri- 
angles, 247;  surface  areas,  249; 
surfaces  of  solids,  252;  contents 
of  solids,  252. 

Metric  System:  linear,  127;  sur- 
face, 131;  solid,  137;  liquid,  141; 
dry,  141 ;  weight,  145. 


288 


CALIFORNIA    SERIES. 


Miscellaneous  Problems,  257. 

Money:  United  States,  1G8;  how 
written,  105;  English,  157; 
French,  157. 

Multiples,  67;  least  common,  G8; 
practical  work  in,  70. 

Multiplication,  34;  analysis  in,  30; 
by  one  figure,  38;  by  lO's,  lOO's, 
etc.,  40;  by  several  figures,  41; 
practical  work  in,  56;  of  frac- 
tions, 82;  of  decimals,  109;  short 
methods,  115;  of  compound 
numbers,  161. 

Notation:  decimal,  5;  Roman,  12; 
of  decimal  fractions,  102. 

Note:  form  of,  214 ;  payable  to  "  or- 
der," 214;  to  "bearer,"  214;  ma- 
turity of,  214;  indorsement  of, 
214;  race  of,  214;  demand  note, 
214 ;  indorsement  on,  215. 

Numbers:  writing  of,  5;  reading  of, 
9;  concrete,  36;  abstract,  36; 
prime,  63;  composite,  63;  inte- 
gral, 72;  mixed,  73;  simple,  122; 
compound,  122. 

Numeration,  9;  names  of  groups, 
9;  of  decimal  fractions,  103. 

Parallelograms,  249;  area  of,  250. 

Partial  Payments,  214,  215. 

Partnership,  178. 

Percentage,  181 

Plastering,  133. 

Polygons,  area  of,  250. 
Powers  and  Roots,  237. 


Present  Worth,  212. 

Profit  and  Loss,  185. 

Proportion:  simple,  176;  com- 
pound, 177. 

Pyramid:  area  of,  253;  contents 
of,  254. 

Ratio,  176. 

Receipt,  form  of,  119. 

Rectangle,  area  of,  129. 

Reduction:  fractions,  74;  com- 
pound numbers,  124. 

Roman  Notation,  12. 

Root:  square,  237;  cube,  241. 

Short  Methods:  in  multiplication, 
115;  in  division,  118. 

Sphere,  252;  surface  of,  254;  vol- 
ume of,  254. 

Stocks,  201. 

Stone  and  Brick  "Work,  137. 

Subtraction,  21;  of  several  figures, 
24;  practical  work  in,  28;  frac- 
tions, 77;  decimals,  108;  com- 
pound numbers,  160. 

Taxes,  198. 

Trapezium,  area  of,  249,  250. 

Trapezoid,  area  of,  249,  250. 

Triangles :  area  of,  250 ;  right  an- 
gle, 247. 

True  Discount,  212. 

Weights:  Avoirdupois,  142;  Troy, 
143;  metric,  145;  apothecaries', 
156 ;  long  ton,  157. 


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